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An Introduction to K-theory

Eric M. Friedlander∗

Department of , Northwestern University, Evanston, USA

Lectures given at the School on Algebraic K-theory and its Applications Trieste, 14 - 25 May 2007

LNS0823001

∗[email protected]

Contents

0 Introduction 5

1 K ( ), K ( ), and K ( ) 7 0 − 1 − 2 − 1.1 Algebraic K0 of rings ...... 7 1.2 Topological K0 ...... 9 1.3 Quasi-projective Varieties ...... 10 1.4 Algebraic vector bundles ...... 12 1.5 Examples of Algebraic Vector Bundles ...... 13 1.6 Picard P ic(X) ...... 14 1.7 K0 of Quasi-projective Varieties ...... 15 1.8 K1 of rings ...... 16 1.9 K2 of rings ...... 17

2 Classifying spaces and higher K-theory 19 2.1 Recollections of theory ...... 19 2.2 BG ...... 20 2.3 Quillen’s ...... 22 2.4 Abelian and exact categories ...... 23 1 2.5 The S− S construction ...... 24 2.6 Simplicial sets and the of a ...... 26 2.7 Quillen’s Q-construction ...... 28

3 Topological K-theory 29 3.1 The Classifying BU Z ...... 29 × 3.2 Bott periodicity ...... 32 3.3 Spectra and Generalized Theories ...... 33 3.4 Skeleta and Postnikov towers ...... 36 3.5 The Atiyah-Hirzebruch Spectral ...... 37 3.6 K-theory Operations ...... 39 3.7 Applications ...... 41

4 Algebraic K-theory and Algebraic 42 4.1 Schemes ...... 42 4.2 Algebraic cycles ...... 44 4.3 Chow Groups ...... 46 4.4 Smooth Varieties ...... 49 4.5 Chern classes and Chern character ...... 51 4.6 Riemann-Roch ...... 53

5 Some Difficult Problems 55 5.1 K (Z) ...... 55 ∗ 5.2 Bass Finiteness ...... 57 5.3 Milnor K-theory ...... 58 5.4 Negative K-groups ...... 59 5.5 Algebraic versus topological vector bundles ...... 60 5.6 K-theory with finite coefficients ...... 60 5.7 Etale K-theory ...... 62 5.8 Integral ...... 63 5.9 K-theory and Quadratic Forms ...... 65

6 Beilinson’s vision partially fulfilled 65 6.1 Motivation ...... 65 6.2 Statement of conjectures ...... 66 6.3 Status of Conjectures ...... 67 6.4 The Meaning of the Conjectures ...... 69 6.5 Etale cohomology ...... 71 6.6 Voevodsky’s sites ...... 74

References 75 An Introduction to K-theory 5

0 Introduction

These notes are a reasonably faithful transcription of lectures which I gave in Trieste in May 2007. My objective was to provide participants of the Al- gebraic K-theory Summer School an overview of various aspects of algebraic K-theory, with the intention of making these lectures accessible to partici- pants with little or no prior knowledge of the subject. Thus, these lectures were intended to be the most elementary as well as the most general of the six lecture series of our summer school. At the of each lecture, various references are given. For example, at the end of Lecture 1 the reader will find references to several very good expositions of aspects of algebraic K-theory which present their subject in much more detail than I have given in these lecture notes. One can view these present notes as a “primer” or a “course outline” which offer a guide to formulations, results, and conjectures of algebraic K-theory found in the literature. The primary topic of each of my six lectures is reflected in the title of each lecture:

1. K ( ), K ( ), and K ( ) 0 − 1 − 2 − 2. Classifying spaces and higher K-theory

3. Topological K-theory

4. Algebraic K-theory and

5. Some Difficult Problems

6. Beilinson’s vision partially fulfilled

Taken together, these lectures emphasize the connections between alge- braic K-theory and algebraic geometry, saying little about connections with and nothing about connections with non-commutative geom- etry. Such omissions, and many others, can be explained by the twin factors of the ignorance of the lecturer and the constraints imposed by the brevity of these lectures. Perhaps what is somewhat novel, especially in such brief format, is the emphasis on the algebraic K-theory of not necessarily affine schemes. Another attribute of these lectures is the continual reference to topological K-theory and algebraic as a source of inspiration and intuition. 6 E.M. Friedlander

We very briefly summarize the content of each of these six lectures. Lec- ture 1 introduces low dimensional K-theory, with emphasis on K0(X), the of finitely generated projective R-modules for a (com- mutative) R if Spec R = X, of topological vector vector bundles over X if X is a finite dimensional C.W. complex, and of coherent, locally free -modules if X is a . Without a doubt, a primary goal (if not the OX primary goal) of K-theory is the understanding of K0. The key concept discussed in Lecture 2 is that of “homotopy theo- retic group completion”, an enriched extension of the process introduced by of taking the group associated to a monoid. We briefly consider three versions of such a group completion, all due to 1 : the plus-construction, the S− S-construction, and the Q- construction. In this lecture, we remind the reader of simplicial sets, abelian categories, and the nerve of a category. The early development of topological K-theory by and Fritz Hirzebruch has been a guide for many algebraic K-theorists during the past 45 years. Lecture 3 presents some of machinery of topological K-theory (spectra in the sense of , the Atiyah-Hirzebruch , and operations in K-theory) which reappear in more recent devel- opments of algebraic K-theory. In Lecture 4 we discuss the relationship of algebraic K-theory to the study of algebraic cycles on (smooth) quasi-projective varieties. In particu- lar, we remind the reader of the definition of Chow groups of algebraic cycles modulo rational equivalence. The relationship between algebraic K-theory and algebraic cycles was realized by Alexander Grothendieck when he first introduced algebraic K-theory; indeed, algebraic K0 figures in the formula- tion of Grothendieck’s Riemann-Roch theorem. As we recall, one beautiful consequence of this theorem is that the Chern character from K0(X) to CH∗(X) of a smooth, quasi- X is a rational equivalence. In order to convince the intrigued reader that there remain many funda- mental questions which await solutions, we discuss in Lecture 5 a few difficult open problems. For example, despite very dramatic progress in recent years, we still do not have a complete computation of the algebraic K-theory of the Z. This lecture concludes somewhat idiosyncratically with a dis- cussion of integral analogues of famous questions formulated in terms of the “semi-topological K-theory” constructed by Mark Walker and the author. The final lecture could serve as an introduction to Professor Weibel’s lectures on the proof of the Bloch-Kato Conjecture. The thread which orga- An Introduction to K-theory 7 nizes the effort of many is a list of 7 conjectures by which proposes to explain to what extent algebraic K-theory pos- sesses properties analogous to those enjoyed by topological K-theory. We briefly discuss the status of these conjectures (all but the Beilinson-Soul´e vanishing conjecture appear to be verified) and discuss briefly the organiza- tional features of the motivic spectral sequence. We conclude this Lecture 6, and thus our series of lectures, with a very cursory discussion of etale cohomology and Grothendieck sites introduced by in his dazzling proof of the Milnor Conjecture.

1 K ( ), K ( ), and K ( ) 0 − 1 − 2 − Perhaps the first new concept that arises in the study of K-theory, and one which recurs frequently, is that of the group completion of an abelian monoid. The basic example to keep in mind is that the of integers Z is the group completion of the monoid N of natural numbers. Recall that an abelian monoid M is a together with a binary, associative, commutative operation + : M M M and a distinguished element 0 M which serves × → ∈ as an identify (i.e., 0 + m = m for all m M). Then we define the group ∈ completion γ : M M + by setting M + equal to the quotient of the free → abelian group with generators [m], m M modulo the generated ∈ by elements of the form [m] + [n] [m + n] and define γ : M M + by − → sending m M to [m]. We frequently refer to M + as the Grothendieck ∈ group of M. The group completion γ : M M + satisfies the following universal → property. For any φ : M A from M to a group A, there → exists a unique homomorphism φ+ : M + A such that φ+ γ = φ : M A. → ◦ →

1.1 Algebraic K0 of rings This leads almost immediately to K-theory. Let R be a ring (always assumed associative with unit, but not necessarily commutative). Recall that an (always assumed left) R- P is said to be projective if there exists another R-module Q such that P Q is a free R-module. ⊕ Definition 1.1. Let (R) denote the abelian monoid (with respect to ) P ⊕ of classes of finitely generated projective R-modules. Then we define K (R) to be (R)+. 0 P 8 E.M. Friedlander

Warning: The group completion map γ : (R) K (R) is frequently not P → 0 injective.

Exercise 1.2. Verify that if j : R S is a ring homomorphism and if P is → a finitely generated projective R-module, then S P is a finitely generated ⊗R projective S-module. Using the of the Grothendieck group, you should also check that this construction determines j : K0(R) K0(S). ∗ → Indeed, we see that K ( ) is a (covariant) from rings to abelian 0 − groups.

Example 1.3. If R = F is a field, then a finitely generated F -module is just a finite dimensional F -. Two such vector spaces are isomorphic if and only if they have the same . Thus, (F ) N and K0(F ) = Z. P ( Example 1.4. Let K/Q be a finite field extension of the rational numbers (K is said to be a number field) and let K be the ring of algebraic OK ⊂ integers in K. Thus, is the subring of those elements α K which satisfy O ∈ a monic pα(x) Z[x]. Recall that K is a . ∈ O The theory of Dedekind domains permits us to conclude that

K0( K ) = Z Cl(K) O ⊕ where Cl(K) is the class group of K.

A well-known theorem of Minkowski asserts that Cl(K) is finite for any number field K (cf. [5]). Computing class groups is devilishly difficult. We do know that there only finitely many cyclotomic fields (i.e., of the form Q(ζn) obtained by adjoining a primitive n-th root of unity to Q) with class group 1 . The smallest n with non-trivial class group is n = 23 { } for which Cl(Q(ζ23)) = Z/3. A check of tables shows, for example, that Cl(Q(ζ100)) = Z/65. The reader is referred to the book [4] for an accessible introduction to this important topic. The K-theory of integral group rings has several important applications in topology. For a group π, the integral group ring Z[π] is defined to be the ring whose underlying abelian group is the on the set [g], g π ∈ and whose ring structure is defined by setting [g] [h] = [g h]. Thus, if π is · · not abelian, then Z[π] is not a . Application 1.5. Let X be a - with the homotopy type of a C.W. complex and with π. Suppose that X is a An Introduction to K-theory 9 retract of a finite C.W. complex. Then the Wall finiteness obstruction is an element of K0(Z[π]) which vanishes if and only if X is homotopy equivalent to a finite C.W. complex.

1.2 Topological K0 We now consider topological K-theory for a X. This is also constructed as a Grothendieck group and is typically easier to compute than algebraic K-theory of a ring R. Moreover, results first proved for topological K-theory have both motivated and helped to prove important theorems in algebraic K-theory. A good introduction to topological K-theory can be found in [1].

Definition 1.6. Let F denote either the real numbers R or the complex numbers C. An F-vector on a topological space X is a continuous open surjective map p : E X satisfying → 1 (a) For all x X, p− (x) is a finite dimensional F-vector space. ∈ (b) There are continuous maps E E E, F E E which provide the × → × → vector space structure on p 1(x), all x X. − ∈ (c) For all x X, there exists an open neighborhood Ux X, an F-vector ∈ ⊂ space V , and a ψ : V U p 1(U ) over U (i.e., x × x → − x x pr = p ψ : V U U ) compatible with the structure in (b). 2 ◦ x × x → x Example 1.7. Let X = S1, the . The of the Mo¨bius band M to its equator p : M S1 is a rank 1, real over S1. → 2 2 Let X = S , the 2-. Then the projection p : T 2 S of the S → is a non-trivial vector bundle. Let X = S2, but now view X as the complex projective line, so that points of X can be viewed as complex lines through the origin in C2 (i.e., complex subspaces of C2 of dimension 1). Then there is a natural rank 1, complex E X whose fibre above x X is the complex line → ∈ parametrized by x; if E o(X) X denotes the result of removing the − → origin of each fibre, then we can identify E o(X) with C2 0 . − − { }

Definition 1.8. Let V ectF(X) denote the abelian monoid (with respect to ) of isomorphism classes of F-vector bundles of X. We define ⊕ 0 + 0 + Ktop(X) = V ectC(X) , KOtop(X) = V ectR(X) . 10 E.M. Friedlander

(This definition will agree with our more sophisticated definition of topo- logical K-theory given in a later lecture provided that the X has the homo- topy type of a finite dimensional C.W. complex.) The reason we use a superscript 0 rather than a subscript 0 for topological K-theory is that it determines a contravariant functor. Namely, if f : X Y → is a continuous map of topological spaces and if p : E Y is an F-vector → bundle on Y , then pr2 : E Y X X is an F-vector bundle on X. This × → determines 0 0 f ∗ : K (Y ) K (X). top → top

Example 1.9. Let nS2 denote the “trivial” rank n, real vector bundle over 2 n 2 2 2 S (i.e., pr2 : R S S ) and let TS2 denote the tangent bundle of S . × → 2 0 2 Then T 2 1 2 3 2 . We conclude that V ect (S ) K (S ) is not S ⊕ S ( S R → Otop injective in this case.

Here is one of the early theorems of K-theory, a theorem proved by Richard Swan. You can find a full proof, for example, in [5].

Theorem 1.10. (Swan) Let F = R (respectively, = C), let X be a compact Hausdorff space, and let (X, F) denote the ring of continuous functions C X F. For any E V ect (X), define the F-vector space of global sections → ∈ F Γ(X, E) to be

Γ(X, E) = s : X E continuous; p s = id . { → ◦ X } Then sending E to Γ(X, E) determines

0 0 KO (X) K0( (X, R)), K (X) K0( (X, C)). top → C top → C 1.3 Quasi-projective Varieties We briefly recall a few basic notions of classical algebraic geometry; a good basic reference is the first chapter of [3]. Let us assume our ground field k is algebraically closed, so that we need only consider k-rational points. For more general fields k, we could have to consider “points with values in some finite field extension L/k.”

Recollection 1.11. Recall PN , whose k-rational points are equivalence classes of N + 1-tuple, a , . . . , a , some entry of which is * 0 N + non-zero. Two N + 1-tuples (a0, . . . , aN ), (b0, . . . , bN ) are equivalent if there exists some 0 = c k such that (a , . . . , a ) = (cb , . . . , cb ). , ∈ 0 N 0 N An Introduction to K-theory 11

If F (X0, . . . , XN ) is a , then the zero locus Z(F ) PN is well defined. ⊂ N N Recall that P is covered by standard affine opens Ui = P Z(Xi). \ Recall the on PN , a of open sets for which are the N of the form UG = P Z(G). \ Recollection 1.12. Recall the notion of a presheaf on a topological space T : a contravariant functor from the category whose objects are open subsets of T and whose are inclusions. Recall that a is a presheaf satisfying the sheaf : for T com- pact, this axiom can be simply expressed as requiring for each pair of open subsets U, V that

F (U V ) = F (U) F (U V ) F (V ). ∪ × ∩ N Recall the structure sheaf of “regular functions” PN on P , sections of P (X0O,...,XN ) N (U) on any open U are given by quotients of homogeneous OP Q(X0,...,XN ) of the same degree satisfying the condition that Q has no zeros in U. In particular,

j N (U ) = F (X)/G , j 0; F homgeneous of deg = j deg(G) . OP G { ≥ · } Definition 1.13. A projective variety X is a space with a sheaf of commu- N N tative rings X which admits a closed into some P , i : X P , O ⊂ so that is the quotient of the sheaf N by the of those regular OX OP functions which vanish on X. A quasi-projective variety U is once again a space with a sheaf of com- N mutative rings U which admits locally a closed embedding into some P , O j : U PN , so that the U PN of U admits the structure of a ⊂ ⊂ projective variety and so that equals the restriction of to U U. OU OU ⊂ A quasi-projective variety U is said to be affine if U admits a closed N N embedding into some A = P Z(X0) so that U is the quotient of N \ O OA by the sheaf of ideals which vanish on U.

Remark 1.14. Any quasi-projective variety U has a base of (Zariski) open subsets which are affine. Most quasi-projective varieties are neither projective nor affine. There is a bijective correspondence between affine varieties and finitely generated commutative k-. If U is an affine variety, then Γ( ) is OU the corresponding finitely generated k-. Conversely, if A is written 12 E.M. Friedlander

N as a quotient k[x1, . . . , xN ] A, then SpecA Spec(k[x1, . . . , xN ]) = A → → is the corresponding closed embedding of the affine variety SpecA.

Example 1.15. Let F be a polynomial in variables X0, . . . , XN homoge- d neous of degree d (i.e., F (ca0, . . . , caN ) = c F (a0, . . . , aN ). Then the zero locus Z(F ) PN is called a of degree d. For example if N = 2, ⊂ then Z(F ) is 1-dimensional (i.e., a curve). If k = C and if the Jacobian of F does not vanish anywhere on C = Z(F ) (i.e., if C is smooth), then C is a (d 1)(d 2) projective, smooth, of − 2 − .

1.4 Algebraic vector bundles Definition 1.16. Let X be a quasi-projective variety. A quasi- on X is a sheaf of -modules (i.e., an abelian sheaf equipped with F OX a pairing of sheaves satisfying the condition that for each open OX ⊗F → F U X this pairing gives (U) the structure of an (U)-module) with the ⊂ F OX property that there exists an open covering U X; i I by affine open { i ⊂ ∈ } subsets so that U is the sheaf associated to an OX (Ui)-module Mi for each F| i i. If each of the Mi can be chosen to be finitely generated as an OX (Ui)- module, then such a quasi-coherent sheaf is called coherent.

Definition 1.17. Let X be a quasi-projective variety. A coherent sheaf E on X is said to be an algebraic vector bundle if is locally free. In other E words, if there exists a (Zariski) open covering Ui; i I of X such that ei { ∈ } Ui X U for each i. E| ( O | i Remark 1.18. If a quasi-projective variety is affine, then an algebraic vector bundle on X is equivalent to a projective Γ( )-module. OX Construction 1. If M is a free A-module of rank r, then the symmetric algebra SymA• (M) is a polynomial algebra of r generators over A and the r structure map π : Spec Sym• (M) Spec A is just the projection A A → × Spec A Spec A. This construction readily globalizes: if is an algebraic → E vector bundle over X, then

π : V( ) Spec Sym• ( )∗ X E E ≡ OX E → is locally in the Zariski topology a projection: if U X; i is { i ⊂ ∈} an open covering restricted to which is trivial, then the restriction of π E r E above each Ui is isomorphic to the product projection A Ui Ui. In × → An Introduction to K-theory 13 the above definition of π we consider the symmetric algebra on the E ∗ = Hom X ( , X ), so that the association V( ∗) is covariantly E O E O E 0→ E functorial. Thus, we may alternatively think of an algebraic vector bundle on X as a map of varieties π : V( ∗) X E E → satisfying properties which are the algebraic analogues of the properties of the structure map of a topological vector bundle over a topological space. Remark 1.19. We should be looking at the of a variety over a field k, rather than simply the k rational points, whenever k is not algebraically closed. We suppress this , for we will soon switch to spectra (i.e., work with schemes of finite type over k). However, we do point out that the reason it suffices to consider the maximal ideal spectrum rather the spectrum of all prime ideals is the validity of the Hilbert Nullstellensatz. One form of this important theorem is that the of maximal ideals constitute a dense subset of the space of prime ideals (with the Zariski topology) of a finitely generated commutative k-algebra.

1.5 Examples of Algebraic Vector Bundles

N Example 1.20. Rank 1 vector bundles OPN (k), k Z on P . The sections N ∈ of O N (j) on the basic open subset UG = P Z(G) are given by the formula P \

OPN (k)(UG) = k[X0, . . . , XN , 1/G](j) (i.e., ratios of homogeneous polynomials of total degree j). In terms of the trivialization on the open covering Ui, 0 i N, the j j ≤ ≤ patching functions are given by Xi /Xi . N+j " Γ(OPN (j)) has dimension j if j > 0, dimension 1 if j = 0, and 0 otherwise. Thus, using the fact that O N (j) O N (j$) = O N (j + j$), ! " P ⊗OX P P we conclude that Γ(OPN (j)) is not isomorphic to Γ(OPN (j$)) provided that j = j. $ , Proposition 1.21. (Grothendieck) Each vector bundle on P1 has a unique decomposition as a finite of copies of 1 (k), k Z. OP ∈ Example 1.22. Serre’s conjecture (proved by Quillen and Suslin) asserts that every algebraic vector bundle on AN (or any affine open subset of AN ) is trivial. In more algebraic terms, every finitely generated projective k[x1, . . . , xn]-module is free. 14 E.M. Friedlander

Example 1.23. Let X = Grassn,N , the Grassmann variety of n 1-planes N N+1 − in P (i.e., n-dimensional subspaces of k ). We can embed Grassn,N as M 1 N+1 a Zariski closed subset of P − , where M = n , by sending the subspace V kN+1 to its n-th power ΛnV Λn(kN+1). There is a natural ⊂ ⊂! " rank n algebraic vector bundle on X provided with an embedding in the E N+1 trivial rank N + 1 dimensional vector bundle X (in the special case N+1 O n = 1, this is N ( 1) ) whose fibre above a point in X is the OP − ⊂ OPN corresponding subspace. Of equal importance is the natural rank N n- − dimensional quotient bundle = N+1/ . Q OPN E This example readily generalizes to flag varieties.

Example 1.24. Let A be a commutative k-algebra and recall the module ¨ ΩA/k of Kahler differentials. These globalize to a quasi-coherent sheaf ΩX on a quasi-projective variety X over k. If X is smooth of dimension r, then ΩX is an algebraic vector bundle over X of rank r.

1.6 P ic(X) Definition 1.25. Let X be a quasi-projective variety. We define P ic(X) to be the abelian group whose elements are isomorphism classes of rank 1 algebraic vector bundles on X (also called “invertible sheaves”). The group structure on P ic(X) is given by product. So defined, P ic(X) is a generalization of the construction of the Class Group (of fractional ideals modulo principal ideal) for X = Spec A with A a Dedekind domain.

Example 1.26. By examining patching data, we readily verify that

1 H (X, ∗ ) = P ic(X) OX where is the sheaf of abelian groups on X with sections Γ(U, ) defined OX∗ OX∗ to be the invertible elements of Γ(U, ) (with group structure given by OX multiplication). If k = C, then we have a short of analytic sheaves of abelian sheaves on the analytic space X(C)an,

exp 0 Z X ∗ 0. → → O → OX → We use identification due to Serre of analytic and algebraic vector bundles on a projective variety. If X = C is a smooth curve, this identification enables An Introduction to K-theory 15 us to conclude the short exact sequence

g 2g 2 0 C /Z P ic(C) H (C, Z) → → → 1 0 g since H (C, C ) H (C, ΩC ) = C (where g is the genus of C). In par- O ( ticular, we conclude that for a curve of positive genus, P ic(C) is very large, having a “continuous part” (which is an ). Example 1.27. A K3 S over the complex numbers is characterized 0 2 1 by the conditions that H (S, Λ (ΩS)) = 0 = Hsing(S, Q). Even though the homotopy type of a smooth does not depend upon the choice of such a surface S, the rank of P ic(S) can vary from 1 to 20. [The dimension 2 of Hsing(S, Q) is 22.]

1.7 K0 of Quasi-projective Varieties

Definition 1.28. Let X be a quasi-projective variety. We define K0(X) to be the quotient of the generated by isomorphism classes [ ] of (algebraic) vector bundles on X modulo the E E generated pairs ([ ], [ ]+[ ]) for each short exact sequence 0 E E1 E2 → E1 → E → 0 of vector bundles. E2 → Remark 1.29. Let A be a finitely generated k-algebra. Observe that every short exact sequence of projective A-modules splits. Thus, the equivalence relation defining K (A) is generated by pairs ([ ], [ ] + [ ]). Every 0 E1 ⊕ E2 E1 E2 element of K (A) can be written as [P ] [m] for some non-negative m; 0 − moreover, projective modules P, Q determine the same class in K0(A) if and only if there exists some non-negative integer m such that P Am Q Am. ⊕ ( ⊕ N Proposition 1.30. K0(P ) is a free abelian group of rank N + 1. More- over, for any k Z, the invertible sheaves PN (k), . . . , PN (k +N) generate N ∈ O O K0(P ). Proof. One obtains a relation among N+2 invertible sheaves from the on N + 1 dimensional vector space V :

N+1 N 1 1 0 Λ V S∗− − (V ) V S∗− (V ) S∗(V ) k 0. → ⊗ → · · · → ⊗ → → → N One shows that the invertible sheaves N (j), j Z generate K0(P ) OP ∈ using Serre’s theorem that for any coherent sheaf on PN there exist integers F n m, n > 0 and a surjective map of PN -modules PN (m) . O N O → F One way to show that the rank of K0(P ) equals N + 1 is to use Chern classes. 16 E.M. Friedlander

1.8 K1 of rings So far, we have only considered degree 0 algebraic and topological K-theory. n Before we consider Kn(R), n N, K (X), n Z, we look explicitly at ∈ top ∈ K1(R). This was first investigated in in the classic book by Bass [2].

Definition 1.31. Let R be a ring (assumed associative, as always and with unit). We define K1(R) by the formula

K (R) GL(R)/[GL(R), GL(R)], 1 ≡ where GL(R) = lim GL(n, R) and where [GL(r), GL(R)] is the commutator n subgroup of the g−r→oup GL(R). Thus, K1(R) is the maximal abelian quotient of GL(R), K1(R) = H1(GL(R), Z).

The commutator subgroup [GL(R), GL(R)] equals the subgroup E(R) ⊂ GL(R) defined as the subgroup generated by elementary matrices E (r), r i,j, ∈ R, i = j (i.e., matrices which differ by the identity matrix by having r in , the (i, j) position). This group is readily seen to be perfect (i.e., E(R) = [E(R), E(R)]); indeed, it is an elementary exercise to verify that E(n, R) = E(R) GL(n, R) is perfect for n 3. ∩ ≥ Proposition 1.32. If R is a commutative ring, then the map

det : K (R) R× 1 → from K1(R) to the multiplicative group of units of R provides a natural split- ting of R = GL(1, R) GL(R) K (R). Thus, we can write × → → 1

K (R) = R× SL(R) 1 × where SL(R) = ker det . { } If R is a field or more generally a , then SK1(R) = 0.

The following theorem is not at all easy, but it does tell us that nothing surprising happens for rings of integers in number fields.

Theorem 1.33. (Bass-Milnor-Serre) If is the ring of integers in a num- OK ber field K, then SK ( ) = 0. 1 OK An Introduction to K-theory 17

Application 1.34. The work of Bass-Milnor-Serre was dedicated to solving the following question: is every subgroup H SL( ) of finite index a ⊂ OK “congruent subgroup” (i.e., of the form ker SL( ) SL( /pn) for { OK → OK } some prime ideal p . The answer is yes if the number field F admits ⊂ OK a real embedding, and no otherwise. The Bass-Milnor-Serre theorem is complemented by the following classi- cal result due to Dirichlet (cf. [5]). Theorem 1.35. (Dirichlet’s Theorem) Let be the ring of integers in a OK number field K. Then

r1+r2 1 O∗ = µ(K) Z − K ⊕ where µ(K) K denotes the finite subgroup of roots of unity and where ⊂ r1 (respectively, r2) denotes the number of of K into R (resp., number of conjugate pairs of embeddings of K into C).

We conclude this brief commentary on K1 with the following early ap- plication to topology. Application 1.36. Let π be a finitely generated group and consider the Whitehead group W h(π) = K (R)/ g; g π . 1 {± ∈ } A homotopy equivalence of finite complexes with fundamental group π has an (its “”) in W h(π) which determines whether or not this is a simple homotopy equivalence (given by a chain of “elementary expansions” and “elementary collapses”).

The interested reader can find a wealth of information about K0 and K1 in the books [2] and [6].

1.9 K2 of rings

One can think of K0(R) as the “stable group” of projective modules “mod- ulo trivial projective modules” and of K1(R) of the stabilized group of auto- morphisms of the trivial projective module modulo “trivial automorphisms” (i.e., the elementary matrices up to isomorphism. This philosophy can be extended to the definition of K2, but has not been extended to Ki, i > 2. Namely, K2(R) can be viewed as the relations among the trivial automor- phisms (i.e., elementary matrices) modulo those relations which hold uni- versally. 18 E.M. Friedlander

Definition 1.37. Let St(R), the Steinberg group of R, denote the group generated by elements X (r), i = j, r R subject to the following commu- i,j , ∈ tator relations: 1 if j = k, i = ' , , [X (r), X (s)] = X (rs) if j = k, i = ' i,j k,"  i," ,  X ( rs) if j = k, i = ' k,j − , We define K (R) to be the kernel of the map St(R) E(R), given by 2  → sending Xi,j(r) to the elementary matrix Ei,j(r), so that we have a short exact sequence 1 K (R) St(R) E(R) 1. → 2 → → → Proposition 1.38. The short exact sequence 1 K (R) St(R) E(R) 1 → 2 → → → is the universal central extension of the perfect group E(R). Thus, K2(R) = H2(E(R), Z), the Schur multiplier of E(R). Proof. One can show that a universal central extension of a group E exists if and only E is perfect. In this case, a group S mapping onto E is the universal central extension if and only if S is also perfect and H2(S, Z) = 0.

Example 1.39. If R is a field, then K1(F ) = F ×, the non-zero elements of the field viewed as an abelian group under multiplication. By a theorem of Matsumoto, K2(F ) is characterized as the target of the “universal Stein- berg symbol”. Namely, K2(F ) is isomorphic to the free abelian group with generators “Steinberg symbols” a, b , a, b F and relations { } ∈ × i. ac,b = a,b c,b , { } { } { } ii. a,bd = a,b a,d , { } { } { } iii. a, 1 a = 1, a = 1 = 1 a. (Steinberg relation) { − } , , − 1 a Observe that for a F ×, a = 1 −a 1 , so that ∈ − − − 1 1 1 1 1 1 a, a = a, 1 a a, 1 a− − = a, 1 a− − = a− , 1 a− = 1. { − } { − }{ − } { − } { − } Then we conclude the skew symmetry of these symbols: a, b b, a = a, a a, b b, a b, b = a, ab b, ab = ab, ab = 1. { }{ } { − }{ }{ }{ − } { − }{ − } { − } Milnor used this presentation of K2(F ) as the starting point of his defini- tion of the Milnor K-theory KMilnor of a field F , discussed briefly in Lecture 5. ∗ An Introduction to K-theory 19

2 Classifying spaces and higher K-theory

2.1 Recollections of Much of our discussions will require some basics of homotopy theory. Two standard references are [8] and [14].

Definition 2.1. Two continuous maps f, g : X Y between topological → spaces are said to be homotopic if there exists some continuous map F : X I Y with F X 0 = f, F X 1 = g (where I denotes the unit × → | ×{ } | ×{ } [0, 1]). If x X, y Y are chosen (“base points”), then two (“pointed”) maps ∈ ∈ f, g : (X, x ) (Y, y ) are said to be homotopic if there exists some { } → { } continuous map F : X I Y such that F X 0 = f, F X 1 = g, and × → | ×{ } | ×{ } F x I = y (i.e., F must project x I to y . We use the notation |{ }× { } { } × { } [(X, x), (Y, y)] to denote the pointed homotopy classes of maps from (X, x) (previously denoted (X, x )) to (Y, y ). { } { } We shall employ the usual notation, [X, Y ] to denote homotopy classes of continuous maps from X to Y . Another basic definition is that of the homotopy groups of a topological space.

Definition 2.2. For any n 0 and any (X, x), ≥ π (X, x) [(Sn, ), (X, x)]. n ≡ ∞ For n = 0, π (X, x) is a ; for n 1, a group; for n 2, an abelian n ≥ ≥ group. If (X, x) is “nice”, then π (X, x) [Sn, X]; moreover, if X is path n ( connected, then the isomorphism class of π (X, x) is independent of x X. n ∈ A relative C.W. complex is a (X, A) (i.e., A is a subspace of X) such that there exists a sequence of subspaces A = X 1 − ⊂ X X of X with equal to X such that X is obtained 0 ⊂ · · · ⊂ n ⊂ · · · n from Xn 1 by “attaching” n-cells (i.e., possibly infinitely many copies of the − closed unit in Rn, where “attachment” means that the of the n 1 disk is identified with its under a continuous map S − Xn 1 ) and → − such that a subset F X is closed if and only if X X X is closed for ⊂ ∩ n ⊂ n all n. A space X is a C.W. complex if (X, ) is a relative C.W. complex. A ∅ pointed C.W. complex (X, x) is a relative C.W. complex for (X, x ). { } C.W. complexes have many good properties, one of which is the following. 20 E.M. Friedlander

Theorem 2.3. () If f : X Y is a continuous map → of connected C.W. complexes such that f : πn(X, x) πn(Y, f(x)) is an ∗ → isomorphism for all n 1, then f is a homotopy equivalence. ≥ Moreover, C.W. complexes are quite general: If (T, t) is a pointed topo- logical space, then there exists a pointed C.W. complex (X, x) and a con- tinuous map g : (X, x) (T, t) such that g : π (X, x) π (T, t) is an → ∗ ∗ → ∗ isomorphism. Recall that a continuous map f : X Y is said to be a fibration if it has → the homotopy : given any commutative square of continuous maps A 0 / X × { }

  A I / Y × then there exits a map A I X whose restriction to A 0 is the upper × → × { } horizontal map and whose composition with the right vertical map equals the lower horizontal map. A very important property of fibrations is that if f : X Y is a fibration, then there is a long exact sequence of homotopy → groups for any x X, y Y : o ∈ ∈ 1 1 πn(f − (y), x0) πn(X, x0) πn(Y, y0) πn 1(f − (y), x0) · · · → → → → − → · · · If f : (X, x) (Y, y) is any pointed map of spaces, we can naturally → construct a fibration f˜ : X˜ Y together with a homotopy equivalence → X X˜ over Y . We denote by htyfib(f) the fibre f˜ 1(y) of f˜. → − 2.2 BG Definition 2.4. Let G be a and X a topological space. Then a G-torsor over X (or principal G-bundle) is a continuous map p : E X together with a continuous action of G on E over X such that there → exists an open covering Ui of X G Ui E U for each { } × → | i i respecting G-actions (where G acts on G U by left multiplication on G). × i Example 2.5. Assume that G is a . Then a G-torsor p : E → X is a normal with G.

Theorem 2.6. (Milnor) Let G be a topological group with the homotopy type of a C.W. complex. There there exists a connected C.W. complex BG and a An Introduction to K-theory 21

G-torsor π : EG BG such that sending a continuous X BG → → to the G-torsor X EG X over X determines a 1-1 correspondence ×BG → [X, BG] & isom classes of G-torsors over X → { } Moreover, the homotopy type of BG is thereby determined; furthermore, EG is contractible.

The topology on G when considering the BG is cru- cial. One interesting construction one can consider is the map on classifying spaces induced by the continuous, bijective function Gδ G where G is a → topological group and Gδ is the same group but provided with the discrete topology.

Corollary 2.7. If G is discrete, then π (BG, ) = G and π (BG, ) = 0 for 1 ∗ n ∗ all n > 0 (where is some choice of base point). Moreover, these properties ∗ characterize the C.W. complex BG up to homotopy type.

n Proof. Sketch of proof. If n > 0, then the facts that π1(S ) = 0 and EG is contractible imply that [Sn, BG] = 0 . The fact that π (BG, ) = G is { } 1 ∗ classical covering space theory.

The proof of the following proposition is fairly elementary, using a stan- dard projection of Z as a Z[π]-module.

Proposition 2.8. Let π be a discrete group and let A be a Z[π]-module. Then H∗(Bπ, A) = Ext∗ (Z, A) H∗(π, A) Z[π] ≡ [π] H (Bπ, A) = T orZ (Z, A) H (π, A). ∗ ∗ ≡ ∗ Now, vector bundles are not G-torsors but rather fibre bundles for the topological groups O(n) (respectively, U(n)) in the case of a real (resp., complex) vector bundle of rank n. Nevertheless, because O(n) (resp., U(n)) acts faithfully and transitively on Rn (resp., Cn), we can readily conclude using Theorem 2.6

[X, BO(n)] = isom classes of real rank n vector bundles over X { } [X, BU(n)] = isom classes of complex rank n vector bundles over X . { } 22 E.M. Friedlander

2.3 Quillen’s plus construction

Daniel Quillen’s original definition of Ki(R), i > 0, was in terms of the follow- ing “Quillen plus construction”. A detailed exposition of this construction can be found in [7]. Theorem 2.9. (Plus construction) Let G be a discrete group and H G ⊂ be a perfect normal subgroup. Then there exists a C.W. complex BG+ and a continuous map γ : BG BG+ → + ˜ such that ker π1(BG) π1(BG ) = H and such that H (htyfib(γ), Z) = { → } ∗ 0. Moreover, γ is unique up to homotopy. The classical “Whitehead Lemma” implies that the commutator sub- group [GL(R), GL(R)] of GL(R) is perfect. (One verifies that an n n × elementary matrix is itself a commutator of elementary matrices provided that n 4.) ≥ Definition 2.10. For any ring R, let

γ : BGL(R) BGL(R)+ → denote the Quillen plus construction with respect to [GL(R), GL(R)] ⊂ GL(R). We define

K (R) π (BGL(R)+), i > 0. i ≡ i This construction is closely connected to the group completions of our

first lecture. In some sense, n BGL(n, R) is “up to homotopy, a commuta- tive topological monoid” and BGL(R)+ Z is a group completion in an ap- ' × propriate sense. There are several technologies which have been introduced 1 in part to justify this informal description (e.g., the “S− S construction” discussed below).

Remark 2.11. Essentially by definition, K1(R) as defined in the first lecture agrees with that of Definition 2.10. Moreover, for any K1(R)-module A,

+ H∗(BGL(R) , A) = H∗(BGL(R), A).

Moreover, one can verify that K2(R) as introduced in the first lecture agrees with that of Definition 2.10 for any ring R by identifying this sec- ond with the second group of the perfect group [GL(R), GL(R)]. An Introduction to K-theory 23

When Quillen formulated his definition of K (R), he also made the fol- ∗ lowing fundamental computation. Indeed, this computation was a motivat- ing factor for Quillen’s definition (cf. [10]).

Theorem 2.12. (Quillen’s computation for finite fields) Let Fq be a finite + field. Then the space BGL(Fq) can be described as the homotopy fibre of a computable map. This leads to the following computation for i > 0:

j Ki(Fq) = Z/q 1 if i = 2j 1 − − Ki(Fq) = 0 if i = 2j. As you probably know, homotopy groups are notoriously hard to com- pute. So Quillen has played a nasty trick on us, giving us very interesting invariants with which we struggle to make the most basic calculations. For example, a fundamental problem which is still not fully solved is to compute Ki(Z). Early computations of higher K-groups of a ring R often proceeded by first computing the group homology groups of GL(n, R) for n large, then relating these homology groups to the homotopy groups of BGL(R)+.

2.4 Abelian and exact categories Much of our discussion in these lectures will require the language and con- cepts of . Indeed, working with categories will give us a method to consider various kinds of K-theories simultaneously. I shall assume that you are familiar with the notion of an abelian cate- gory. Recall that in an , the set of morphisms Hom (B, C) A A for any A, B Obj has the natural structure of an abelian group; more- ∈ A over, for each A, B Obj , there is an object B C which is both a ∈ A ⊕ product and a ; moreover, any f : A B in Hom (A, B) has → A both a and a cokernel. In an abelian category, we can work with exact just as we do in the category of abelian groups. Example 2.13. Here are a few “standard” examples of abelian categories.

the category Mod(R) of (left) R-modules. • the category mod(R) of finitely generated R-modules (in which case • we must take R to be ).

the category QCoh(X) of quasi-coherent sheaves on a variety X. • 24 E.M. Friedlander

the category Coh(X) of coherent sheaves on a Notherian variety X. •

Warning. The full subcategory (R) mod(R) is not an abelian category. P ⊂ For example, if R = Z, then n : Z Z is a homomorphism of projective → R-modules whose cokernel is not projective and thus is not in (Z). P Definition 2.14. An is a full additive subcategory of some P abelian category such that A (a) There exists some set S Obj such that every A Obj is isomorphic ⊂ A ∈ A to some element of S. (b) If 0 A A A 0 is an exact sequence in with both → 1 → 2 → 3 → A A , A Obj , then A Obj . 1 3 ∈ P 2 ∈ P An admissible monomorphism (respectively, epimorphism) in is a mono- P A A (resp., A A ) in which fits in an exact sequence 1 → 2 2 → 3 P of the form of (b).

Definition 2.15. If is an exact category, we define K ( ) to be the group P 0 P completion of the abelian monoid defined as the quotient of the monoid of isomorphism classes of objects of (with respect to ) modulo the equiva- P ⊕ lence relation [A ] [A ] [A ] for every exact sequence of the form (I.5.b). 2 − 1 − 3 Exercise 2.16. Show that K (R) equals K ( (R)), where (R) is the exact 0 0 P P category of finitely generated projective R-modules. More generally, show that K (X) equals K ( ect(X)), where ect(X) is 0 0 V V the exact category of algebraic vector bundles on the quasi-projective variety X.

Definition 2.17. Let be an exact category in which all exact sequences P split. Consider pairs (A, α) where A Obj and α is an automorphism of ∈ P A. Direct sums and exact sequences of such pairs are defined in the obvious way. Then K ( ) is defined to be the group completion of the abelian 1 P monoid defined as the quotient of the monoid of isomorphism classes of such pairs modulo the relations given by short exact sequences.

1 2.5 The S− S construction Recall that a symmetric monoidal category S is a (small) category with a unit object e S and a functor : S S S which is associative and com- ∈ ! × → mutative up to coherent natural isomorphisms. For example, if we consider An Introduction to K-theory 25 the category of finitely generated projective R-modules, then the direct P sum : is associative but only commutative up to natural iso- ⊕ P × P → P morphism. The symmetric monoidal category relevant for the K-theory of a ring R is the category Iso( ) whose objects are finitely generated projec- P tive R-modules and whose morphisms are isomorphisms between projective R-modules. 1 Quillen’s construction of S− S for a symmetric monoidal category S is appealing, modelling one way we would introduce inverses to form the group completion of an abelian monoid. A good reference for this is [13].

Definition 2.18. Let S be a symmetric monoidal category. The category S 1S is the category whose objects are pairs a, b of objects of S and − { } whose maps from a, b to c, d are equivalence classes of compositions of { } { } the following form:

s (f,g) a, b !− s a, s b) c, d { } → { ! ! → { } where s is some object of S, f, g are morphisms in S. Another such compo- sition s"! (f ",g") a, b − s$ a, s$ b) c, d { } → { ! ! → { } is declared to be the same map in S 1S from a, b to c, d if and only − { } { } if there exists some isomorphism θ : s s such that f = f (θ a), g = → $ $ ◦ ! g (θ b). $ ◦ ! Heuristically, we view a, b S 1S as representing a b, so that { } ∈ − − s!a, s!b also represents a b. Moreover, we are forcing morphisms in { } 1 − S to be invertible in S− S. If we were to apply this construction to the nat- ural numbers N viewed as a discrete category with addition as the operation, 1 then we get N− N = Z. 1 The following theorem of Quillen shows how the S− S construction can provide a homotopy-theoretic group completion

Theorem 2.19. (Quillen) Let S be a symmetric monoidal category with the property that for all s, t S the map s : Aut(t) Aut(s t) is injective. ∈ !− → ! Then the natural map BS B(S 1S) of classifying spaces (see the next → − ) is a homotopy-theoretic group completion. In particular, if S denotes the category whose objects are finite dimen- sional projective R-modules and whose maps are isomorphisms (so that BS = BAut(P )), then (R) is homotopy equivalent to B(S 1S). [P ] K − ' 26 E.M. Friedlander

2.6 Simplicial sets and the Nerve of a Category The reader is referred to [9] for a detailed introduction to simplicial sets.

Definition 2.20. The category of standard simplices, ∆, has objects n = 0, 1, . . . , n indexed by n N and morphisms given by * + ∈ Hom (m, n) = non-decreasing maps 0, 1, . . . , n 0, 1, . . . , m . ∆ { * + → * +} The special morphisms

∂ : n-1 n (skip i); σ : n+1 n (repeat j) i → j → in ∆ generate (under composition) all the morphisms of ∆ and satisfy certain standard relations which many topologists know by heart. A S is a functor ∆op (sets). • →

In other words, S consists of a set Sn for each n 0 and maps di : Sn • ≥ → Sn 1, sj : Sn Sn+1 satisfying the relations given by the relations satisfied − → by ∂ , σ ∆. i j ∈ Example 2.21. Let T be a topological space. Then the singular complex Sing T is a simplicial set. Recall that SingnT is the set of continuous • maps ∆n T , where ∆n Rn+1 is the standard n-: the subspace → ⊂ consisting of those points x = (x , . . . , x ) with each x 0 and x = 1. 0 n i ≥ i Since any map µ : n m determines a (linear) map ∆n ∆m, it also → → ( determines µ : Sing T Sing T , so that we may easily verify that m → n Sing T : ∆op (sets) • → is a well-defined functor.

Definition 2.22. (Milnor’s geometric realization functor) For any simplicial set X , we define its geometric realization as the topological space X. given • | | as follows: n X = Xn ∆ / | •| × ∼ n 0 )≥ where the equivalence relation is given by (x, µ t) (µ x, t) whenever n ◦ ( ◦ x Xm, t ∆ , µ : n m a map of ∆. This quotient is given the quotient ∈ ∈ → n topology, where each Xn ∆ is topologized as a disjoint union indexed by n × n+1 x Xn of copies of ∆ R . ∈ ⊂ An Introduction to K-theory 27

Now, simplicial sets are a very good combinatorial model for homotopy theory as the next theorem reveals.

Theorem 2.23. () The categories of topological spaces and simplicial sets satisfy the following relationships.

Milnor’s geometric realization functor is left adjoint to the singular • functor; in other words, for every simplicial set X and every topolog- • ical space T ,

Hom(s.sets)(X , Sing T ) = Hom(spaces)( X , T ). • • | •|

For any simplicial set X , X is a C.W. complex; moreover, for any • • | •| topological space T , Sing. (T ) is a particularly well behaved type of • simplicial set called a Kan complex.

For any topological space T and any point t T , the adjunction mor- • ∈ phism ( Sing T , t) (T, t) | • | → induces an isomorphism on homotopy groups.

The adjunction morphisms above induce an equivalence of categories • (Kan cxes)/ hom.equiv (C.W. cxes)/ hom.equiv . ∼ ( ∼

Now we can define the classifying space of a (small) category.

Definition 2.24. Let be a small category. We define the nerve N C C ∈ (s.sets) to be the simplicial set whose set of n-simplices is the set of com- posable n-tuples of morphisms in : C γn γ1 N n = Cn Cn 1 C0 . C { → − → · · · → }

For ∂i : n-1 n, we define di : N n N n 1 to send the n-tuple Cn → C → C − → C0 to that n 1-tuple given by composing γi+1 and γi whenever · · · → − γ1 γn 0 < i < n, by dropping C if i = 0 and by dropping C if i = n. For → 0 n → σ : n n+1, we define s : N N by repeating C and inserting j → j Cn → Cn+1 j the identity map. We define the classifying space B of the category to be N , the C C | C| geometric realization of the nerve of . C 28 E.M. Friedlander

The reader is encouraged to consult [12] for a discussion and insight into this construction.

Example 2.25. Let G be a (discrete) group and let denote the category G with a single object (denoted ) and with Hom ( , ) = G. Then B is a ∗ G ∗ ∗ G model for BG (i.e., B is a connected C.W. complex with π (B , ) = G G 1 G ∗ and all higher homotopy groups 0).

Example 2.26. Let X be a and let (X) denote the category S whose objects are simplices of X and maps are the inclusions of simplices. Then B (X) can be identified with the first barycentric subdivision of X. S

2.7 Quillen’s Q-construction

What are the higher K-groups of an exact category? In particular, what are the higher K-groups of a quasi-projective variety X (i.e., of the exact category ect(X)) or more generally of a scheme? V Quillen defines these in terms of another construction, the “Quillen Q- construction.” This construction as well as many fundamental applications can be found in Quillen’s remarkable paper [11].

Definition 2.27. Let be an exact category and let Q be the category ob- P P tained from by applying the Quillen Q-construction (as discussed below). P Then K ( ) = π (BQ ), i 0, i P i+1 P ≥ the homotopy groups of the geometric realization of the nerve of Q . P Theorem 2.28. Let X be a scheme and let ect(X) denote the exact cate- V gory of finitely presented, locally free -modules. Then OX K (X) π ( ect(X)) π (BQ ect(X)) i ≡ i V ≡ i+1 V agrees for i = 0 with the Grothendieck group of ect(X) and for X = SpecA V + an affine scheme agrees with Ki(A) = πi(BLG(A) ) provided that i > 0.

1 Quillen proves this theorem using the S− S construction as an interme- diary. Here is the formulation of Quillen’s Q-construction. An Introduction to K-theory 29

Definition 2.29. Let be an exact category. We define the category Q P P as follows. We set Obj Q equal to Obj . For any A, B Obj Q , we P P ∈ P define

p i HomQ (A, B) = A " X # B; p (resp. i) admissible epi (resp. mono)/ P { ∼} where the equivalence relation is generated by pairs

A " X # B, A " X$ # B which fit in a commutative diagram

i A o p X / B = =   i  o $ / A p X B $ Waldhausen in [15] gives a somewhat more elaborate construction of Quillen’s Q construction which produces “n-fold deloopings” of BQ for P every n 0: pointed spaces T with the property that Ωn(T ) is homotopy ≥ n n equivalent to BQ . P

3 Topological K-theory

In this lecture, we will discuss some of the machinery which makes topological K-theory both useful and computable. Not only does topological K-theory play a very important role in topology, but also it has played the most important guiding role in the development of algebraic K-theory. As general references, the books [17], [18] and [14] are recommended.

3.1 The Classifying space BU Z × The following statements about topological vector bundles are not valid (in general) for algebraic vector bundles. These properties suggest that topo- logical K-theory is better behaved than algebraic K-theory.

Proposition 3.1. (cf. [1]) Let T be a compact Hausdorff space. If p : E T is a topological vector bundle on T , then for some N > 0 there is a → surjective map of bundles on T , (CN+1 T ) E. × → 30 E.M. Friedlander

Any surjective map E F of topological vector bundles on T admits a → splitting over T . The set of homotopy classes of maps [T, BU(n)] is in natural 1-1 corre- spondence with the set of isomorphism classes of rank n topological vector bundles on T .

Proof. The first statement is proved using a partition of unity argument. The proof of the second statement is by establishing a Hermitian on E (so that E F F ), which is achieved by once again using a partition ( ⊕ ⊥ of unity argument. To prove the last statement, one verifies that if T I G is a homotopy × → relating continuous maps f, g : T G and if E is a topological vector bundle → on G, then f E g E as topological vector bundles on T . Once again, a ∗ ( ∗ partition of unity argument is the key ingredient in the proof.

Proposition 3.2. For any space T , the set of homotopy classes of maps

[T, BU Z], BU = lim BUn × −→n admits a natural structure of an abelian group induced by block sum of ma- trices U U U . We define n × m → n+m 0 K (T ) [T, BU Z]. top ≡ × 0 For any compact, Hausdorff space T , Ktop(T ) is naturally isomorphic to the Grothendieck group of topological vector bundles on T :

Z[iso classes of top vector bundles on T ] K0 (T ) . top ( [E] = [E ] + [E ], whenever E E E 1 2 ( 1 ⊕ 2 Proof. (External) direct sum of matrices gives a monoid structure on BU 7n n which determines a (homotopy associative and commutative) H-space struc- ture on BU Z which we view as the mapping telescope of the self map × BU BU , BU + BU BU . 7n n → 7n n i × { ∈ 1} → i+1 The (abelian) group structure on [T, BU Z] is then determined. × To show that this mapping telescope is actually an H-space, one must verify that it has a 2-sided identity up to pointed homotopy: one must verify that product on the left with + BU1 gives a self map of BU Z which ∈ × is related to the identity via a base-point preserving homotopy. (Such a An Introduction to K-theory 31 verification is not difficult, but the analogous verification fails if we replace the topological groups Un by discrete groups GLn(A) for some unital ring A.)

Example 3.3. Since the Lie groups Un are connected, the spaces BUn are simply connected and thus

0 1 K (S ) = π1(BU Z) = 0. top × It is useful to extend K0 ( ) to a relative theory which applies to pairs top − (T, A) of spaces (i.e., T is a topological space and A T is a closed subset). ⊂ In the special case that A = , then T/ = T /+, the pointed space obtained ∅ ∅ + by taking the disjoint union of T with a point + which we declare to be the basepoint.

Definition 3.4. If T is a pointed space with basepoint t0, we define the reduced K-theory of T by

K˜ ∗ (T ) K∗ (T, t ). top ≡ top 0 For any pair (T, A), we define

K0 (T, A) K˜ 0 (T/A) top ≡ top 0 thereby extending our earlier definition of Ktop(T ). For any n > 0, we define

Kn (T, A) K˜ 0 (Σn(T/A)). top ≡ top In particular, for any n 0, we define ≥ n n 0 n K− (T ) K− (T, ) K˜ (Σ (T )). top ≡ top ∅ ≡ top + Observe that

K˜ 0 (S T ) = ker K0 (S T ) K0 (S) K0 (T ) , top ∧ { top × → top ⊕ top } so that (external) of bundles induces a natural pairing

i j i j K− (S) K− (T ) K− − (S T ). top ⊗ top → top × 32 E.M. Friedlander

Just to get the notation somewhat straight, let us take T to be a single point T = t . Then T+ = t, + , the 2-point space with new point + as { } 2 { } 2 base-point. Then Σ (T+) is the 2-sphere S , and thus

2 0 2 0 K− ( t ) = ker K (S )) K (+) . top { } { top → top } We single out a special element, the Bott element

2 β = [ 1 (1)] [ 1 ] K− (pt)), OP − OP ∈ top where we have abused notation by identifying (P1)an with S2 and the images 1 0 1 an of algebraic vector bundles on P in Ktop((P ) ) have the same names as in 1 K0(P ).

3.2 Bott periodicity Of fundamental importance in the study of topological K-theory is the fol- lowing theorem of . Recall that if (X, x) is pointed space, then the space ΩX is the function complex (with the compact-open topol- ogy) of continuous maps from (S1, ) to (X, x). The functor ∞ Ω( ) on pointed spaces is adjoint to the functor Σ( ): there is − − a natural

Maps(Σ(X), Y ) Maps(X, Ω(Y )) ( of sets of continuous, pointed (i.e, base point preserving) maps. An extensive discussion of Bott periodicity can be found in [17].

Theorem 3.5. (Bott Periodicity) There are the following homotopy equiv- alences.

From BO Z to its 8-fold loop space: • × 8 BO Z Ω (BO Z) × ∼ ×

Moreover, the homotopy groups πi(BO Z) are given by × Z, Z/2, Z/2, 0, Z, 0, 0, 0

depending upon whether i is congruent to 0, 1, 2, 3, 4, 5, 6, 7 modulo 8. An Introduction to K-theory 33

From BU Z to its 2-fold loop space: • × 2 BU Z Ω (BU Z) × ∼ ×

Moreover, πi(BU Z) is Z if i is even and equals 0 if i is odd. × Atiyah interprets this 2-fold periodicity in terms of K-theory as follows.

Theorem 3.6. (Bott Periodicity) For any space T and any i 0, multipli- ≥ cation by the Bott element induces a natural isomorphism

i i 2 β : K− (T ) K− − (T ). top → top i Using the above theorem, we define Ktop(X) for any topological space X i and any integer i as Ktop(X), where i is 0 if i is even and i is -1 if i is odd. ˜ 0 2 In particular, taking T to be a point, we conclude that Ktop(S ) = Z, generated by the Bott element.

Example 3.7. Let S0 denote , + = . According to our definitions, {∗ } ∗+ the K-theory K ( ), of a point equals the reduced K-theory of S0. In top ∗ particular, for n > 0,

n n 0 0 n K− ( ) = K˜ − (S ) = K˜ (S ) = π (BU). top ∗ top top n Thus, we conclude

n Z if n is even Ktop( ) = ∗ * 0 if n is odd We can reformulate this by writing

i n Z if i + n is even Ktop(S ) = * 0 if i + n is odd

3.3 Spectra and Generalized Cohomology Theories

Thus, both BO Z and BU Z are “infinite loop spaces” naturally deter- × × mining Ω-spectra in the following sense.

Definition 3.8. A spectrum E is a of pointed spaces E0, E1, . . . , each { } of which has the homotopy type of a pointed C.W. complex, together with continuous structure maps Σ(Ei) Ei+1. → 34 E.M. Friedlander

The spectrum E is said to be an Ω-spectrum if the adjoint Ei Ω(Ei+1) → of each map is a homotopy equivalence; in other words, a sequence of pointed homotopy equivalences

E0 & ΩE1 & Ω2E2 & & ΩnEn → → → · · · → → · · · Each spectrum E determines an Ω-spectrum E˜ defined by setting

j j n E˜n = lim Ω Σ − (En). −→j The importance of Ω-spectra is clear from the following theorem which asserts that an Ω-spectrum determines a “generalized cohomology theory”. Theorem 3.9. (cf. [14]) Let E be an Ω-spectrum. For any topological space X with closed subspace A X, set ⊂ hn (X, A) = [(X, A), En], n 0 E ≥ Then (X, a) h (X, A) is a generalized cohomology theory; namely, this 0→ E∗ satisfies all of the Eilenberg-Steenrod except that its value at a point (i.e., ( , )) may not be that of ordinary cohomology: ∗ ∅ (a) h ( ) is a functor from the category of pairs of spaces to graded abelian E∗ − groups. (b) for each n 0 and each pair of spaces (X, A), there is a functorial ≥ connecting homomorphism ∂ : hn (A) hn+1(X, A). E → E (c) the connecting of (b) determine long exact sequences for every pair (X, A). (d) h ( ) satisfies excision: i.e., for every pair (X, A) and every subspace E∗ − U A whose closure lies in the of A, h (X, A) h (X U, A U). ⊂ E∗ ( E∗ − − Observe that in the above definition we use the notation hE∗ (X) for h (X, ) = h (X , ), where X is the disjoint union of X and a point . E∗ ∅ E∗ + ∗ + ∗ Definition 3.10. The (periodic) topological K-theories KO ( ), K ( ) t∗op − t∗op − are the generalized cohomology theories associated to the Ω-spectra given by BO Z and BU Z with their deloopings given by Bott periodicity. × × In particular, whenever X is a finite dimensional C.W. complex, 2j 2j 1 K (X) = [X, BU Z], K − (X) = [X, U], top × top 0 0 so that we recover our definition of Ktop(X) (and similarly KOtop(X)). An Introduction to K-theory 35

Let us restrict attention to Kt∗op(X) which suffices to motivate our further discussion in algebraic K-theory. (K0t∗op(X) motivates Hermetian algebraic K-theory.) There are also other interesting generalized cohomology theories (e.g., theory represented by the infinite loop space MU) which play a role in algebraic K-theory, and there are also more sophisticated equivariant K-theories, none of which will we discuss in these lectures. Tensor product of vector bundles induces a multiplication

K0 (X) K0 (X) K0 (X) top ⊗ top → top for any finite dimensional C.W. complex X. This can be generalized by ob- serving that tensor product induces group homomorphisms U(m) U(n) × → U(n + m) and thereby maps of classifying spaces

BU(m) BU(n) BU(n + m). × → With a little effort, one can show that these multiplication maps are compati- ble up to homotopy with the standard embeddings U(m) U(m+1), U(n) ⊂ ⊂ U(n + 1) and thereby give us a pairing

(BU Z) (BU Z) BU Z × × × → × (factoring through the ). In this way, BU Z has the structure × of an H-space which induces a pairing of spectra and thus a multiplication for the generalized cohomology theory K ( ). (A completely similar argument t∗op − applies to KO ( )). t∗op − i Remark: Each of the topological K-groups, Kt−op(X), i N, is given as 0 i i th ∈ Ktop(Σ X) where Σ X is the i suspension of X. On the other hand, alge- braic K-groups in non-zero degree are not easily related to the algebraic K0 of some associated ring. As an example of how topological K-theory inspired even the early (very algebraic) effort in algebraic K-theory we mention the following classical theorem of Hyman Bass. The analogous result in topological K-theory for rank e vector bundles over a finite dimension C.W. complex of dimension d < e can be readily proved using the standard method of “”. Theorem 3.11. (Bass stability theorem) Let A be a commutative, noethe- rian ring of d. Then for any two projective A-modules P, P $ of rank e > d, if [P ] = [P ] K (A) then P must be isomorphic to P . $ ∈ 0 $ 36 E.M. Friedlander

3.4 Skeleta and Postnikov towers

If X is a C.W. complex then we can define its p-skeleton skp(X) for each p 0 as the subspace of X consisting of the union of those cells of dimension ≥ p. Thus, the C.W. complex can be written as the union (or colimit) of its ≤ skeleta, X = sk (X). ∪p p There is a standard way to “chop off” the bottom homotopy groups of a space (or an Ω-spectrum) using an analogue of the universal covering space of a space (which “chops off” the fundamental group).

Definition 3.12. Let X be a C.W. complex. For each n 0, construct ≥ a map X X[n] by attaching cells (proceeding by dimension) to kill all → homotopy groups of X above dimension n 1. Define − X(n) to X, htyfib X X[n] . { → } So defined, X(n) X induces an isomorphism on homotopy groups π , i n → i ≥ and π (X(n)) = 0, j n. j ≤ The Postinov tower of X is the sequence of spaces

X X(n+1) X(n) · · · → → → · · · Thus, X can be viewed as the “homotopy inverse ” of its Postnivkov tower. Algebraic K-theory corresponds most closely the topological K-theory which is obtained by replacing the Ω-spectrum K = BU Z by kU = bu Z × × obtained by taking at stage i the ith connected of BU Z starting at × stage 0. The associated generalized cohomology theory is denoted kU ( ) ∗ − and satisfies kU i(X) Ki (X), i 0. ( top ≤

In studying the mapping complex Mapcont(X, Y ) continuous maps from a C.W. complex X to a space Y , one typically filters this mapping complex using the skeleton filtration of X by its skeleta or the “coskeleton” filtration of Y by its Postnikov tower. We refer to [14] for details of these complemen- tary approaches. An Introduction to K-theory 37

3.5 The Atiyah-Hirzebruch Spectral sequence The Atiyah-Hirzebruch spectral sequence for topological K-theory has been a strong motivating factor in recent developments in algebraic K-theory. Indeed, perhaps the fundamental criterion for is that it should satisfy a relationship to algebraic K-theory strictly analogous to the relationship of singular cohomology to topological K-theory. Theorem 3.13. (Atiyah-Hirzebruch spectral sequence [16]) For any gener- alized cohomology theory h ( ) and any topological space X, there exists a E∗ − right half- spectral sequence of cohomological type

Ep,q = Hp(X, hq( )) hp+q(X). 2 ∗ ⇒ E

The filtration on hE∗ (X) is given by p F E∗ = ker h∗ (X) h∗ (skp(X) . ∞ { E → E } In the special case of K ( ), this takes the following form t∗op − p,q p p+q E = H (X, Z(q/2)) K (X) 2 ⇒ top where Z(q/2) = Z if q is even and 0 otherwise. In the special case of kU ( ), this takes the following form ∗ − p,q p p+q E = H (X, Z(q/2)) kU (X) 2 ⇒ where Z(q/2) = Z if q is an even non-positive integer and 0 otherwise. Proof. There are two basic approaches to proving this spectral sequence. The first is to assume T is a cell complex, then consider T as a filtered space with T T the union of cells of dimension n. The properties of K ( ) n ⊂ ≤ t∗op − stated in the previous theorem give us an exact couple associated to the long exact sequences

q n q q q Ktop(S ) Ktop(Tn/Tn 1) Ktop(Tn) Ktop(Tn 1) · · · → ⊕ ( − → → − → Kq+1(Sn) ⊕ top → · · · where the direct sum is indexed by the n-cells of T . The second approach applies to a general space T and uses the Postnikov tower of BU Z. This is a tower of fibrations whose fibers are Eilenberg- × MacLane spaces for the groups which occur as the homotopy groups of BU × Z. 38 E.M. Friedlander

What is a spectral sequence of cohomological type? This is the data of p,q a 2-dimensional array Er of abelian groups for each r r (typically, r ≥ 0 0 equals 0, or 1 or 2; in our case r0 = 2) and homomorphisms

p,q p,q p+r,q r+1 d : E E − r r → r p,q such that the next array Er+1 is given by the cohomology of these homo- morphisms: p,q p,q p r,q+r 1 E = ker d /im d − − . r+1 { r } { r } p,q To say that the spectral sequence is “right half plane” is to say Er = 0 whenever p < 0. We say that the spectral sequence converges to the abutment E∗ (in our case h∗ (X)) if at each spot (p, q) there are only ∞ E finitely many non-zero homomorphisms going in and going out and if there exists a decreasing filtration F pEn on each En so that { ∞} ∞ En = F pEn , 0 = F pEn , ∞ ∞ ∞ p p + ,

p n p+1 n p,n p F E /F E = ER − , R >> 0. ∞ ∞

The Postnikov tower argument together with a knowledge of the k- invariants of BU Z shows that after tensoring with Q this Atiyah-Hirzebruch × , , spectral sequence collapses; in other words, that E∗ ∗ Q = E∗ ∗ Q. 2 ⊗ ∞ ⊗ Theorem 3.14. ([16]) Let X be a C.W. complex. Then there are isomor- phisms

0 ev 1 odd kU (X)) Q H (X, Q), kU − (X) Q H (X, Q). ⊗ ( ⊗ ( These isomorphisms are induced by the Chern character

ev ch = chi : K0( ) H ( , Q) − → − -i discussed in Lecture 4. While we are discussing spectral sequences, we should mention the fol- lowing: An Introduction to K-theory 39

Theorem 3.15. (; cf. [14]) Let (B, b) be a connected, pointed C.W. complex. For any fibration p : E B of topological spaces 1 → with fibre F = p− (b) and for any abelian group A, there exists a convergent first quadrant spectral sequence of cohomological type Ep,q = Hp(B, Hq(F, A)) Hp+q(E, A) 2 ⇒ provided that π1(B, b) acts trivially on H∗(F, A). The non-existence of an analogue of the Serre spectral sequence in alge- braic geometry (for cohomology theories based on algebraic cycles or alge- braic K-theory) presents one of the most fundamental challenges to compu- tations of algebraic K-groups.

3.6 K-theory Operations There are several reasons why topological K-theory has sometimes proved to be a more useful computational tool than singular cohomology.

K0 ( ) can be torsion free, even though Hev( , Z) might have torsion. • top − − This is the case, for example, for compact Lie groups.

K∗ ( ) is essentially Z/2-graded rather than graded by the natural • top − numbers. K ( ) has interesting cohomology operations not seen in cohomol- • t∗op − ogy. These operations originate from the observation that the exterior products Λi(P ) of a projective module P are likewise projective mod- ules and the exterior products Λi(E) of a vector bundle E are likewise vector bundles.

A good introduction to K-theory operations can be found in the appendix of [1]. Definition 3.16. Let X be a finite dimensional C.W. complex and E X → be a topological vector bundle of rank r. Define r λ (E) = [ΛiE]ti K0 (X)[t], t ∈ top -i=0 0 a polynomial with constant term 1 and thus an invertible element in Ktop(X)[[t]]. Extend this to a homomorphism 0 0 λ : K (X) (1 + K (X)[[t]])∗, t top → top 40 E.M. Friedlander

i 0 (using the fact that λt(E F ) = λt(E) λt(F )) and define λ : Ktop(T ) 0 ⊕ i · → Ktop(T ) to be the coefficient of t of λt. 0 For a general topological space X, define these λ operations on Ktop(X) for by defining them first on the universal vector bundles over and using the functoriality of K0 ( ). top − In particular, J. introduced operations

ψk( ) : K0 ( ) K0 ( ), k > 0 − top − → top − (called Adams operations) which have many applications and which are similarly constructed for algebraic K-theory.

Definition 3.17. For any topological space T , define

i i d ψt(x) = ψ (X)t rank(x) t (logλ t(x)) ≡ − · dt − i 0 -≥ for any x K0 (T ). ∈ top The Adams operations ψk satisfy many good properties, some of which we list below.

Proposition 3.18. For any topological space T , any x, y K0 (T ), any ∈ top k > 0

ψk(x + y) = ψk(x) + ψk(y). • ψk(xy) = ψk(x)ψk(y). • ψk(ψ"(x) = ψk"(x). • k q 2q chq(ψ (x)) = k chq(x) H (T, Q). • ∈ ψp(x) is congruent modulo p to xp if p is a . • ψk(x) = xk whenever x is a line bundle • In particular, if E is a sum of line bundles L , then ψk(E) = ((L )k), ⊕i i ⊕ i the k-th power sum. By the splitting principle, this property alone uniquely determines ψk. top We introduce further operations, the γ-operations on K0 (T ). An Introduction to K-theory 41

Definition 3.19. For any topological space T , define

i i γt(x) = γ (X)t λt/1 t(x) ≡ − i 0 -≥ for any x K0 (T ). ∈ top Basic properties of these γ-operations include the following

1. γt(x + y) = γt(x)γt(y) 2. γ([L] 1) = 1 + t([L] 1). − −

3. λs(x) = γs/1+s(x) 0 Using these γ operations, we define the γ filtration on Ktop(T ) as follows. γ,1 Definition 3.20. For any topological space T , define Ktop (T ) as the kernel of the rank map

Kγ,1(T ) ker rank : K0 (T ) K0 (π (T )) . top ≡ { top → top 0 } For n > 1, define

K0 (T )γ,n Kγ,0(T ) K0 (T ) top ⊂ top ≡ top to be the subgroup generated by monomials γi1 (x ) γik (x ) with i 1 · · · k j j ≥ n, x Kγ,1(T ). i ∈ top ( 3.7 Applications We can use the Adams operations and the γ-filtration to describe in the 0 following theorem the relationship between Ktop(T ), a group which has no natural grading, and the graded group Hev(T, Q). Theorem 3.21. Let T be a finite cell complex. Then for any k > 0, ψk γ,n restricts to a self-map of each Ktop (T ) and satisfies the property that it induces multiplication by kn on the quotient

ψk(x) = kn x, x Kγ,n(T )/Kγ,n+1(T )). · ∈ top top Furthermore, the Chern character ch induces an isomorphism

γ,n γ,n+1 2n chn : K (T )/K (T )) Q H (T, Q). top top ⊗ ( 42 E.M. Friedlander

In particular, the preceding theorem gives us a K-theoretic way to define the grading on K0 (T ) Q induced by the Chern character isomorphism. top ⊗ The graded piece of (the associated graded of) K0 (T ) Q corresponding top ⊗ to H2n(T, Q) consists of those classes x for which ψk(x) = knx for some (or all) k > 0. Here is a short list of famous theorems of Adams using topological K- theory and Adams operations:

Application 3.22. Adams used his operations in topological K-theory to solve fundamental problems in algebraic topology. Examples include:

Determination of the number of linearly independent vector fields on • the n-sphere Sn for all n > 1.

Determination of the only (namely, n = 1, 2, 4, 8) for which • Rn admits the structure of a . (The examples of the real numbers R, the complex numbers C, the quaternions, and the Cayley numbers gives us structures in these dimensions.)

Determination of those (now well understood) elements of the homo- • 0 n topy groups of associated with KOtop(S ).

4 Algebraic K-theory and Algebraic Geometry

4.1 Schemes Although our primary interest will be in the K-theory of smooth, quasi- projective algebraic varieties, for completeness we briefly recall the more gen- eral context of schemes. (A good basic reference is [3].) A quasi-projective variety corresponds to a globalization of a finitely generated over a field; a scheme similarly corresponds to the globalization of a general commutative ring. Recall that if A is a commutative ring we denote by Spec A the set of prime ideals of A. The set X = SpecA is provided with a topology, the Zariski topology defined as follows: a subset Y X is closed if and only ⊂ if there exists some ideal I A such that Y = p X; I p . We define ⊂ { ∈ ⊂ } the structure sheaf of commutative rings on X = Spec A by specifying OX its value on the basic X = p SpecA, f / p for some f A to f { ∈ ∈ } ∈ be the ring Af obtained from A by adjoining the inverse to f. (Recall that A A sends to 0 any element a A such that f n a = 0 for some n). → f ∈ · An Introduction to K-theory 43

We now use the sheaf axiom to determine the value of on any arbitrary OX open set U X, for any such U is a finite union of basic open subsets. The ⊂ stalk of the structure sheaf at a prime ideal p A is easily computed OX,p ⊂ to be the local ring A = f / p 1A. p { ∈ }− Thus, (X = Spec A, ) has the structure of a local : a OX topological space with a sheaf of commutative rings each of whose stalks is a local ring. A map of local ringed spaces f : (X, ) (Y, ) is the data OX → OY of a continuous map f : X Y of topological spaces and a map of sheaves → 1 OY f OX on Y , where f OX (V ) = OX (f − (V )) for any open V Y . → ∗ ∗ ⊂ If M is an A-module for a commutative ring A, then M defines a sheaf M˜ of -modules on X = Spec A. Namely, for each basic open subset OX X X, we define M˜ (X ) A M. This is easily seen to determine f ⊂ f ≡ f ⊗A a sheaf of abelian groups on X with the additional property that for every open U X, M˜ (U) is a sheaf of (U)-modules with structure compatible ⊂ OX with restriction to smaller open subsets U U. $ ⊂ Definition 4.1. A local ringed space (X, ) is said to be an affine scheme OX if it is isomorphic (as local ringed spaces) to (X = Spec A, ) as defined OX above. A scheme (X, ) is a local ringed space for which there exists OX a finite open covering Ui i I of X such that each (Ui, X U ) is an affine { } ∈ O | i scheme. If k is a field, a k-variety is a scheme (X, ) with the property there OX is a finite open covering Ui i I by affine schemes with the property that { } ∈ each (Ui, X U ) (SpecAi, SpecAi ) with Ai a finitely generated k-algebra O | i ( O without nilpotents. The (Spec A , ) are affine varieties admitting a i OSpec Ai locally closed embedding in PN , where N + 1 is the cardinality of some set of generators of Ai over k. Example 4.2. The scheme 1 is a non-affine scheme defined by patching to- PZ 1 gether two copies of the affine scheme SpecZ[t]. So P has a covering U1, U2 Z { } corresponding to rings A1 = Z[u], A2 = Z[v]. These are “patched together” by identifying the open subschemes Spec(A ) SpecA , Spec(A ) 1 u ⊂ 1 2 v ⊂ SpecA via the isomorphism of rings (A ) (A ) which sends u to v 1. 2 1 u ( 2 v − Note that we have used SpecR to denote the local ringed space (SpecR, ); we will continue to use this abbreviated notation. OSpecR Definition 4.3. Let (X, ) be a scheme. We denote by ect(X) the OX V exact category of sheaves F of -modules with the property that there OX exists an open covering U of X by affine schemes U = SpecA and free, { i} i i 44 E.M. Friedlander

finitely generated Ai-modules Mi such that the restriction F U of F to Ui is | i isomorphic to the sheaf M˜ on SpecA . In other words, ect(X) is the exact i i V category of coherent, locally free -modules (i.e., of vector bundles over OX X). We define the algebraic K-theory of the scheme X by setting K (X) = K ( ect(X)). ∗ ∗ V 4.2 Algebraic cycles For simplicity, we shall typically restrict our attention to quasi-projective varieties. In some sense, the most intrinsic objects associated to an are the (algebraic) vector bundles E X and the algebraic cycles → Z X on X. As we shall see, these are closely related. ⊂ Definition 4.4. Let X be a scheme. An algebraic r-cycle on X if a formal sum nY [Y ], Y irreducible of dimension r, nY Z ∈ -Y with all but finitely many nY equal to 0. Equivalently, an algebraic r-cycle is a finite integer combination of (not necessarily closed) points of X of dimension r. (This is a good definition even for X a quite general scheme.) If Y X is a reduced subscheme each of whose irreducible components ⊂ Y1, . . . , Ym is r-dimensional, then the algebraic r-cycle m Z = [Yi] -i=1 is called the cycle associated to Y . The group of (algebraic) r-cycles on X will be denoted Zr(X). For example, if X is an integral variety of dimension d (i.e., the field of fractions of X has transcendence d over k), then a Weil divisor is an algebraic d 1-cycle. In the following definition, we extend to r-cycles the − equivalence relation we impose on locally principal divisor when we consider these modulo principal divisors. As motivation, observe that if C is a smooth curve and f frac(C), then f determines a morphism f : C P1 and ∈ → 1 1 (f) = f − (0) f − ( ), − ∞ where f 1(0), f 1( ) are the scheme-theoretic fibres of f above 0, . − − ∞ ∞ An Introduction to K-theory 45

Definition 4.5. Two r-cycles Z, Z$ on a quasi-projective variety X are said to be rationally equivalent if there exist algebraic r + 1-cycles W0, . . . , Wn on 1 X P for some n > 0 with the property that each component of each Wi × 1 projects onto an open subvariety of P and that Z = W0[0], Z$ = Wn[ ], and ∞ W [ ] = W [0] for 0 i < n. Here, W [0] (respectively, W [ ] denotes the i ∞ i+1 ≤ i i ∞ cycle associated to the scheme theoretic fibre above 0 P1 (resp., P1) 1 1 ∈ ∞ ∈ of the restriction of the projection X P P to (the components of) Wi. × → The CHr(X) is the group of r-cycles modulo rational equiv- alence.

Observe that in the above definition we can replace the role of r+1-cycles on X P1 and their geometric fibres over 0, by r + 1-cycles on X U × ∞ × for any non-empty Zariski open U X and geometric fibres over any two ⊂ k-rational points p, q U. ∈ Remark 4.6. Given some r + 1 dimensional irreducible subvariety V X ⊂ together with some f k(V ), we may define (f) = ord (f)[S] where S ∈ S S runs through the codimension 1 irreducible subvarieties of V . Here, ord ( ) ( S − is the valuation centered on S if V is regular at the codimension 1 point corresponding to S; more generally, ordS(f) is defined to be the length of the OV,S-module OV,S/(f). We readily check that (f) is rationally equivalent to 0: namely, we as- 1 sociate to (V, f) the closure W = Γf X P of the of the rational ⊂ × map V P1 determined by f. Then (f) = W [0] W [ ]. $$% − ∞ Conversely, given an r+1-dimensional irreducible subvariety W on X P1 pr2 × which maps onto P1, the composition W X P1 P1 determines f ⊂ × → ∈ frac(W ) such that (f) = W [0] W [ ]. − ∞ Thus, the definition of rational equivalence on r-cycles of X can be given in terms of the equivalence relation generated by

(f), f frac(W ); W irreducible of dimension r + 1 { ∈ } In particular, we conclude that the subgroup of principal divisors inside the group of all locally principal divisors consists precisely of those locally principal divisors which are rationally equivalent to 0.

The reader is referred to the beginning of [20] for a discussion of algebraic cycles and equivalence relations on cycles. 46 E.M. Friedlander

4.3 Chow Groups One should view CH (X) as a homology/cohomology theory. These groups ∗ are covariantly functorial for proper maps f : X Y and contravariantly → functorial for flat maps W X, so that they might best be viewed as some → sort of Borel-Moore homology theory.

Construction 1. Assume that X is integral and regular in codimension 1. Let P ic(X) be a locally free sheaf of rank 1 (i.e., a “line bundle” L ∈ or “”) and assume that Γ( ) = 0. Then any 0 = s Γ( ) L , , ∈ L determines a well defined locally principal divisor on X, Z(s) X. Namely, ⊂ if U X U is trivial when restricted to some open U X, then sU L| ( O | ⊂ ∈ (U) determines an element of (U) well defined up to a unit in (U) L OX OX (i.e., an element of X∗ (U)) so that the valuation vx(s) is well defined for (1) O every x U . We define Z(s) by the property that Z(s)U = (sU ) U for ∈ | any open U X restricted to which is trivial, and where (s ) denotes ⊂ L U the divisor of an element of X (U) corresponding to sU under any ( X ) U O O | isomorphism U ( X ) U . L| ( O | Theorem 4.7. (cf. [3]) Assume that X is an integral variety regular in codimension 1. Let (X) denote the group of locally principal divisors on D X modulo principal divisors. Then the above construction determines a well defined isomorphism P ic(X) (X). ( D Moreover, if is a unique factorization domain for every x X, then OX,x ∈ D(X) equals the group CH1(X) of codimension 1 cycles modulo rational equivalence.

Proof. If s, s Γ( ) are non-zero global sections, then there exists some $ ∈ L f K = frac( ) such that with respect to any trivialization of on ∈ OX L some open covering U X of X the quotient of the images of s, s in { i ⊂ } $ (U ) equals f. A line bundle is trivial if and only if it is isomorphic to OX i L which is the case if and only if it has a global section s Γ(X) which OX ∈ never vanishes if and only if (s) = 0. If , are two such line bundles with L1 L2 non-zero global sections s , s , then (s s ) = (s ) + (s ). 1 2 1 ⊗ 2 1 2 Thus, the map is a well defined homomorphism on the monoid of those line bundles with a non-zero global section. By Serre’s theorem concerning coherent sheaves generated by global sections, for any line bundle there L exists a positive integer n such that X (n) is generated by global L ⊗OX O An Introduction to K-theory 47 sections (and in particular, has non-zero global sections), where we have M implicitly chosen a locally closed embedding X P and taken X (n) to ⊂ O be the pull-back via this embedding of M (n). Thus, we can send such an OP P ic(X) to (s) (w), where s Γ( X (n)) and w Γ( X (n)). L ∈ − ∈ L ⊗OX O ∈ O The fact that P ic(X) (X) is an isomorphism is an exercise in un- → D ravelling the formulation of the definition of line bundle in terms of local data. Recall that a domain A is a unique factorization domain if and only every prime of height 1 is principal. Whenever is a unique factorization do- OX,x main for every x X, every codimension 1 subvariety Y X is thus locally ∈ ⊂ principal, so that the natural inclusion D(X) CH1(X) is an equality. ⊂ Remark 4.8. This is a first example of relating bundles to cycles, and moreover a first example of . Namely, P ic(X) is the group of rank 1 vector bundles; the group CH1(X) of is a group of cycles. Moreover, P ic(X) is contravariant with respect X whereas Z1(X) is covariant with respect to equidimensional maps. To relate the two as in the above theorem, some conditions are required.

Example 4.9. Let X = AN . Then any N 1-cycle (i.e., Weil divisor) N − N Z CHN 1(A ) is principal, so that CHN 1(A ) = 0. ∈ − − More generally, consider the map µ : AN A1 PN A1 which sends × → × (x , . . . , x ), t to t x , . . . , t x , 1 , t. Consider an irreducible subvariety 1 n * · 1 · n + Z AN of dimension r > N not containing the origin and Z PN be its ⊂ 1 1 ⊂ closure. Let W = µ− (Z A ). Then W [0] = whereas W [1] = Z. Thus, N × ∅ CHr(A ) = 0 for any r < N. Example 4.10. Arguing in a similar geometric fashion, we see that the N 1 N N 1 inclusion of a linear plane P − P induces an isomorphism CHr(P − ) = N ⊂ CHr(P ) provided that r < N and thus we conclude by induction that N N 1 N 1 CHr(P ) = Z if r N. Namely, consider µ : P A P A sending ≤ × →1 × 1 x0, . . . , xN , t to x), . . . , xN 1, t xN , t and set W = µ− (Z A ) for any * + * − · + × Z not containing 0, . . . , 0, 1 . Then W [0] = prN (Z), W [1] = Z. * + ∗ Example 4.11. Let C be a smooth curve. Then P ic(C) CH (X). ( 0 Definition 4.12. If f : X Y is a proper map of quasi-projective varieties, → then the proper push-forward of cycles determines a well defined homomor- phism f : CHr(X) CHr(Y ), r 0. ∗ → ≥ 48 E.M. Friedlander

Namely, if Z X is an irreducible subvariety of X of dimension r, then [Z] is ⊂ sent to d [f(Z)] CH (Y ) where [k(Z) : k(f(Z))] = d if dim Z = dim f(Z) · ∈ r and is sent to 0 otherwise. If g : W X is a flat map of quasi-projective varieties of relative dimen- → sion e, then the flat pull-back of cycles induces a well defined homomorphism

g∗ : CH (X) CH (W ), r 0. r → r+e ≥ Namely, if Z X is an irreducible subvariety of X of dimension r, then [Z] ⊂ is sent to the cycle on W associated to Z W W . ×X ⊂ Proposition 4.13. Let Y be a closed subvariety of X and let U = X Y . \ Let i : Y X, j : U X be the inclusions. Then the sequence → → i j∗ CH (Y ) ∗ CH (X) CH (U) 0 r → r → r → is exact for any r 0. ≥ Proof. If V U is an irreducible subvariety of U of dimension r, then the ⊂ closure of V in X, V X, is an irreducible subvariety of X of dimension r ⊂ with the property that j∗([V ]) = [V ]. Thus, we have an exact sequence

i j∗ Z (Y ) ∗ Z (X) Z (U) 0. r → r → r → If Z = n [Y ] is a cycle on X with j (Z) = 0 CH (U), then j Z = i i i ∗ ∈ r ∗ (f) where each W U is an irreducible subvarieties of U of dimension W,f ( ⊂ r + 1 and f k(W ). Thus, Z = n [Y ] (f) is an r-cycle on Y ( ∈ $ i i i − W ,f with the property that i (Z$) is rationally equivalent to Z. Exactness of the ∗ ( ( asserted sequence of Chow groups is now clear.

Corollary 4.14. Let H PN be a hypersurface of degree d. Then N ⊂ CHN 1(P H) = Z/dZ. − \ The following “examples” presuppose an understanding of “smoothness” briefly discussed in the next section. Example 4.15. Mumford shows that if S is a projective smooth surface with a non-zero global algebraic 2-form (i.e., H0(S, Λ2(Ω )) = 0), then CH (S) S , 0 is not finite dimensional (i.e., must be very large). Bloch’s Conjecture predicts that if S is a projective, smooth surface with 0 2 geometric genus equal to 0 (i.e., H (S, Λ (ΩS)) = 0), then the natural map from CH0(S) to the (finite dimensional) is injective. An Introduction to K-theory 49

4.4 Smooth Varieties We restrict our attention to quasi-projective varieties over a field k.

Definition 4.16. A quasi-projective variety X is smooth of dimension n at some point x X if there exists an open neighborhood x U X ∈ ∈ ⊂ and k polynomials f1, . . . , fk in n + k variables (viewed as regular functions ∂f on An+k) vanishing at 0 An+k with Jacobian i (0) of rank k and an ∈ | ∂xj | n+k isomorphism of U with Z(f1, . . . , fk) A sending x to 0. ⊂ In more algebraic terms, a point x X is smooth if there exists an open ∈ neighborhood x U X and a map p : U An sending x to 0 which is ∈ ⊂ → flat and unramified at x.

Definition 4.17. Let X be a quasi-projective variety. Then K0$ (X) is the Grothendieck group of isomorphism classes of coherent sheaves on X, where the equivalence relation is generated pairs ([ ], [ ] + [ ]) for short exact E E1 E2 sequences 0 0 of -modules. → E1 → E → E2 → OX Example 4.18. Let A = k[x]/x2. Consider the short exact sequence of A-modules 0 k A k 0 → → → → where k is an A-module via the augmentation map (i.e., x acts as multipli- cation by 0), where the first map sends a k to ax A, and the second ∈ ∈ map sends x to 0. We conclude that the class [A] of the rank 1 equals 2[k]. On the other hand, because A is a local ring, K0(A) = Z, generated by the class [A]. Thus, the natural map K (Spec A) K (Spec A) is not 0 → 0$ surjective. The map is, however, injective, as can be seen by observing that dimk( ) : K$ (Spec A) Z is well defined. − 0 → Theorem 4.19. If X is smooth, then the natural map K (X) K (X) is 0 → 0$ an isomorphism.

Proof. Smoothness implies that every coherent sheaf has a finite resolution by vector bundles, This enables us to define a map

K$ (X) K (X) 0 → 0 by sending a coherent sheaf to the alternating sum ΣN ( 1)i , where F i=1 − Ei 0 0 is a resolution of by vector bundles. → EN → · · · E0 → F → F 50 E.M. Friedlander

Injectivity follows from the observation that the composition

K (X) K$ (X) K (X) 0 → 0 → 0 is the identity. Surjectivity follows from the observation that = ΣN ( 1)i F i=1 − Ei so that the composition

K$ (X) K (X) K$ (X) 0 → 0 → 0 is also the identity.

Perhaps the most important consequence of this is the following obser- vation. Grothendieck explained to us how we can make K ( ) a covariant 0$ − functor with respect to proper maps. (Every morphism between projective varieties is proper.) Consequently, restricted to smooth schemes, K ( ) is 0 − not only a contravariant functor but also a covariant functor for proper maps. “Chow’s Moving Lemma” is used to give a ring structure on CH∗(X) on smooth varieties as made explicit in the following theorem. The role of the moving lemma is to verify for an r-cycle Z on X and an s-cycle W on X that Z can be moved within its rational to some Z$ such that Z$ meets W “properly”. This means that the intersection of any of Z$ with any irreducible component of W is either empty or of codimension d r s, where d = dim(X). − − Theorem 4.20. Let X be a smooth quasi-projective variety of dimension d. Then there exists a pairing

CHr(X) CHs(X) • CHd r s(X), d r + s, ⊗ → − − ≥ with the property that if Z = [Y ], Z$ = [W ] are irreducible cycles of dimen- sion r, s respectively and if Y W has dimension d r s, then Z Z is ∩ ≤ − − • $ a cycle which is a sum with positive coefficients (determined by local data) indexed by the irreducible subvarieties of Y W of dimension d r s. s ∩ − − Write CH (X) for CHd s(X). With this indexing convention, the in- − tersection pairing has the form

CHs(X) CHt(X) • CHs+t(X). ⊗ → Proof. Classically, this was proved by showing the following geometric fact: given a codimension r cycle Z and a codimension s cycle W = j mjRj with r + s d, then there is another codimension r cycle Z = n Y ≤ $ ( i i i ( An Introduction to K-theory 51 rationally equivalent to Z (i.e., determining the same element in CHr(X)) such that Z$ meets W “properly”; in other words, every component Ci,j,k of each Y R has codimension r + s. One then defines i ∩ j

Z$ W = n m int(Y R , C )C • i j · i ∩ j i,j,k i,j,k -i,j,k where int(Y R , C ) is a positive integer determined using local commu- i ∩ j i,j,k tative algebra, the intersection multiplicity. Furthermore, one shows that if one chooses a Z$$ rationally equivalent to both Z, Z$ and also intersecting W properly, then Z W is rationally equivalent to Z” W . $ • • A completely different proof is given by William Fulton and Robert MacPherson (cf. [20]). They use a powerful geometric technique discov- ered by MacPherson called deformation to the normal . For Y X 1 ⊂ closed, the deformation space MY (X) is a variety mapping to P defined as the in the blow-up of X P1 along Y of the blow-up of × × ∞ X along Y . One readily verifies that Y P1 M(X, Y ) restricts × ∞ × ∞ × ⊂ above = p P1 to the given embedding Y X; and above , restricts to ∞ , ∈ ⊂ ∞n n1 the inclusion of Y into the normal cone CY (X) = Spec( n 0 Y / ), where ⊕ ≥ I IY is the ideal sheaf defining Y X. When Y X is a regular closed IY ⊂ OX ⊂ ⊂ embedding, then this normal cone is a bundle, the normal bundle NY (X). This enables a regular closed embedding (e.g., the diagonal δ : X → X X for X smooth) to be deformed into the embedding of the 0-section of × the normal bundle N (X X). One defines the intersection of Z, W as δ(X) × the intersection of δ(X), Z W and thus one reduces the problem of defining × intersection product to the special case of intersection of the 0-section of the normal bundle NX (X X) with the normal cone N(Z W ) δ(X)(Z W ). × × ∩ ×

4.5 Chern classes and Chern character

The following construction of Chern classes is due to Grothendieck (cf. [19]); it applies equally well to topological vector bundles (in which case the Chern classes of a topological vector bundle over a topological space T are elements of the singular cohomology of T ). If is a rank r + 1 vector bundle on a quasi-projective variety X, we E define P( ) = Proj (SymOX ) X to be the of lines in E E → . Then P( ) comes equipped with a canonical line bundle P( )(1); for X E E r O E a point, P( ) = P and ( )(1) = Pr (1). E OP E O 52 E.M. Friedlander

Construction 2. Let be a rank r vector bundle on a smooth, quasi- E projective variety X of dimension d. Then CH∗(P( )) is a free module over 2 r 1 E 1 CH∗(X) with generators 1, ζ, ζ , . . . , ζ − , where ζ CH (P( )) denotes ∈ E the divisor class associated to ( )(1). . OP E We define the i-th c ( ) CHi(X) of by the formula i E ∈ E r i r i CH∗(P( )) = CH∗(X)[ζ]/ ( 1) π∗(ci( )) ζ − . E − E · -i=0 We define the total Chern class c( ) by the formula E r c( ) = c ( ) E i E -i=0 and set c ( ) = r c ( )ti. Then the Whitney sum formula asserts that t E i=0 i E c ( ) = c ( ) c ( ). t E ⊕ F t E (· t F We define the Chern roots, α , . . . , α of by the formula 1 r E r c ( ) = (1 + α t) t E i .i=1 where the factorization can be viewed either as purely formal or as occurring in F( ). Observe that ck( ) is the k-th elementary symmetric function of E E these Chern roots.

In other words, the Chern classes of the rank r vector bundle are E given by the expression for ζr CHr(P( )) in terms of the generators r 1 ∈ E 1, ζ, . . . , ζ − . Thus, the Chern classes depend critically on the identification of the first Chern class ζ of ( )(1) and the multiplicative structure on OP E CH∗(X). The necessary structure for such a definition of Chern classes is called an oriented multiplicative cohomology theory. The splitting principle guarantees that Chern classes are uniquely determined by the assignment of first Chern classes to line bundles. Grothendieck introduced many basic techniques which we now use as a matter of course when working with bundles. The following splitting princi- ple is one such technique, a technique which enable one to frequently reduce constructions for arbitrary vector bundles to those which are a sum of line bundles. An Introduction to K-theory 53

Proposition 4.21. (Splitting Principle) Let be a rank r + 1 vector bundle E on a quasi-projective variety X. Then p1∗ : CH (X) CH +r(P( )) is split ∗ → ∗ E injective and p ( ) = is a direct sum of a rank r bundle and a line bundle. 1∗ E E1 Applying this construction to 1 on P( ), we obtain p2 : P( 1) P( ); E E E → E proceeding inductively, we obtain

p = pr p1 : F( ) = P( r 1) X ◦ · · · ◦ E E − → with the property that p∗ : K0(X) K0(F( )) is split injective and p∗( ) is → E E a direct sum of line bundles.

One application of the preceding proposition is the following definition (due to Grothendieck) of the Chern character.

Construction 3. Let X be a smooth, quasi-projective variety, let be a E rank r vector bundle over X, and let π : F( ) X be the associated bundle E → of flags of . Write π∗( ) = 1 r, where each i is a line bundle on E E r L ⊕ · · · ⊕ L L F( ). Then ct(π∗( )) = (1 c1( i))t. E E i=1 ⊕ L We define the Chern character of as / E r r 1 1 ch( ) = 1 + c ( ) + c ( )2 + c ( )3 + = exp(c ( )), E { 1 Li 2 1 Li 3! 1 Li · · · } t Li -i=1 -i=1 where this expression is verified to lie in the image of the injective map CH∗(X) Q CH∗(F( )) Q. (Namely, one can identify chk( ) as the ⊗ → E ⊗ E k-th power sum of the Chern roots, and therefore expressible in terms of the Chern classes using Newton polynomials.)

Since π∗ : K0(X) K0(F( )), π∗ : CH∗(X) CH∗(F( )) are ring → E → E homomorphisms, the splitting principle enables us to immediately verify that ch is also a ring homomorphism (i.e., sends the direct sum of bundles to the sum in CH∗(X) of Chern characters, sends the tensor product of bundles to the product in CH∗(X) of Chern characters).

4.6 Riemann-Roch Grothendieck’s formulation of the Riemann-Roch theorem is an assertion of the behaviour of the Chern character ch with respect to push-forward maps induced by a proper smooth map f : X Y of smooth varieties. It is → not the case that ch commutes with the these push-forward maps; one must 54 E.M. Friedlander modify the push forward map in K-theory by multiplication by the . This modification by multiplication by the Todd class is necessary even when consideration of the push-forward of a divisor. Indeed, the Todd class

td : K (X) CH∗(X) 0 → is characterized by the properties that

i. td(L) = c (L)/(1 exp( c (L)) = 1 + 1 c (L) + ; 1 − − 1 2 1 · · · ii. td(E E ) = td(E ) td(E ); and 1 ⊕ 2 1 · 2 iii. td f = f td. ◦ ∗ ∗ ◦ The reader is recommended to consult [19] for an excellent exposition of Grothendieck’s Riemann-Roch Theorem.

Theorem 4.22. (Grothendieck’s Riemann-Roch Theorem) Let f : X Y be a projective map of smooth varieties. Then for any → x K (X), we have the equality ∈ 0

ch(f!(x)) td(TY ) = f (ch(x) td(TX )) · ∗ · where TX , TY are the tangent bundles of X, Y and td(TX ), td(TY ) are their Todd classes. Here, f : K (X) K (Y ) is defined by identifying K (X) with K (X), ! 0 → 0 0 0$ K0(Y ) with K0$ (Y ), and defining f! : K0$ (X) K0$ (Y ) by sending a coherent i i → sheaf on X to i( 1) R f (F ). The map f : CH (X) CH (Y ) is F − ∗ ∗ ∗ → ∗ proper push-forward of cycles. ( Just to make this more concrete and more familiar, let us consider a very special case in which X is a projective, smooth curve, Y is a point, and x K (X) is the class of a line bundle . (Hirzebruch had earlier proved a ∈ 0 L version of Grothendieck’s theorem in which the target Y was a point.)

Example 4.23. Let C be a projective, smooth curve of genus g and let f : C SpecC be the projection to a point. Let be a line bundle on C → L with first Chern class D CH1(C). Then ∈ 1 f!([ ]) = dim (C) dimH (C, ) Z, L L − L ∈ An Introduction to K-theory 55 and ch : K0(SpecC) = Z A∗(SpecC) = Z is an isomorphism. Let K → ∈ CH1(C) denote the “canonical divisor”, the first Chern class of the line bundle ΩC , the dual of TC . Then K 1 td(TC ) = − = 1 K. 1 (1 + K + 1 K2) − 2 − 2 Recall that deg(K) = 2g 2. Since ch([ ]) = 1 + D, we conclude that − L 1 1 f (ch([ ]) td(TC )) = f ((1 + D) (1 K)) = deg(D) deg(K). ∗ L · ∗ · − 2 − 2 Thus, in this case, Riemann-Roch tell us that

dim (C) dimH1(C, ) = deg(D) + 1 g. L − L − For our purpose, Riemann-Roch is especially important because of the following consequence.

Corollary 4.24. Let X be a smooth quasi-projective variety. Then

ch : K0(X) Q CH∗(X) Q ⊗ → ⊗ is a ring isomorphism.

Proof. The essential ingredient is the Riemann-Roch theorem. Namely, we have a natural map CH (X) K (X) sending an irreducible subvariety W ∗ → 0$ to the -module . We show that the composition with the Chern char- OX OW acter is an isomorphism by applying Riemann-Roch to the closed W W X W . \ sing → \ sing 5 Some Difficult Problems

As we discuss in this lecture, many of the basic problems formulated years ago for algebraic K-theory remain unsolved. This remains a subject in which much exciting work remains to be done.

5.1 K (Z) ∗ Unfortunately, there are few examples (rings or varieties) for which a com- plete computation of the K-groups is known. As we have seen earlier, one such complete computation is the K-theory of an arbitrary finite field, 56 E.M. Friedlander

K (Fq). Indeed, general theorems of Quillen give us the complete computa- ∗ tions

1 K (Fq[t]) = K (Fq), K (Fq([t, t− ]) = K (Fq) K 1(Fq). ∗ ∗ ∗ ∗ ⊕ ∗− Perhaps the first natural question which comes to mind is the following: “what is the K-theory of the integers.”

In recent years, great advances have been made in computing K ( K ) ∗ O of a ring of integers in a number field K (e.g., Z inside Q).

K0( K ) Q is 1 dimensional by the finiteness of the class number of • O ⊗ K (Minkowski).

K1( K ) Q has dimension r1 + r2 1, where r1, r2 are the numbers • O ⊗ − of real and complex embeddings of K. (Dirichlet).

Quillen proved that K ( ) is a finitely generated abelian group for • i OK any i.

For i > 1, Borel determined •

0, i 0 (mod 4) ≡ r1 + r2, i 1 (mod 4) Ki( K ) Q =  ≡ (1) O ⊗ 0, i 2 (mod 4)  ≡ r2, i 3 (mod 4)  ≡   in terms of the numbers r1, r2.

K (OK , Z/2) has been computed by Rognes-Weibel as a corollary of • ∗ Voevodsky’s proof of the Milnor Conjecture.

K (Z, Z/p) follows in all degrees not divisible by 4 from the Bloch-Kato • ∗ Conjecture, now seemingly proved by Rost and Voevodsky.

Here is a table of the values of K (Z) whose likely inaccuracy is due to ∗ my confusion of indexing of Bernoulli numbers. Many more details can be found in [27]. An Introduction to K-theory 57

Theorem 5.1. The K-theory of Z is given by (according to Weibel’s survey paper):

K8k = ?0?, 0 < k K8k+1 = Z Z/2, 0 < k  ⊕ K8k+2 = Z/2c2k+1 Z/2  ⊕  K8k+3 = Z/2d4k+2, i 3  ≡ (2) K8k+4 = ?0?  K8k+5 = Z  K8k+6 = Z/c2k+2   K8k+7 = Z/d4k+4   Here, ck/dk is definedto be the reduced expression for Bk/4k, where Bk is the k-th Bernoulli number (defined by

t ∞ B = 1 + k t2k . et 1 (2k)! − -k=1

Challenge 5.2. Prove the vanishing of K4i(Z), i > 0.

5.2 Bass Finiteness Conjecture This is one of the most fundamental and oldest conjectures in algebraic K- theory. Very little progress has been made on this in the past 35 years.

Conjecture 5.3. (Bass finiteness) Let A be a commutative ring which is

finitely generated as an algebra over Z. Is Kn$ (A) (i.e., the Quillen K-theory of mod(A)) finitely generated for all n? In particular, if A is regular as well as commutative and finitely generated over Z, is each Kn(A) a finitely generated abelian group?

This conjecture seems to be very difficult, even for n = 0. There are similar finiteness conjectures for the K-theory of projective varieties over finite fields.

Example 5.4. Here is an example of Bass showing that we must assume A is regular or consider G (A). Let A = Z[x, y]/x2. Then the ideal (x) ∗ is infinitely additively generated by x, xy, xy2, . . . . On the other hand, if t (x), then 1 + t A , so that we see that K (A) is not finitely generated. ∈ ∈ ∗ 1 58 E.M. Friedlander

Example 5.5. As pointed out by Bass, it is elementary to show (using general theorems of Quillen and Quillen’s computation of the K-theory of finite fields) that if A is finite, then G (A) G (A/radA) is finite for every n ( n n 0. Subsequently, Kuku proved that K (A) is also finite whenever A is ≥ n finite (see [32]).

There are many other finiteness conjectures involving smooth schemes of finite type over a finite field, Z or Q. Even partial solutions to these conjectures would represent great progress.

5.3 Milnor K-theory We recall Milnor K-theory, a major concept in Professor Vishik’s lectures. This theory is motivated by Matsumoto’s presentation of K2(F ) (mentioned in Lecture 1),

Definition 5.6. (Milnor) Let F be a field with multiplicative group of units Milnor F ×. The Milnor K-group Kn (F ) is defined to be the n-th graded piece n of the defined as the tensor algebra n 0(F ×)⊗ modulo the ideal generated by elements a, 1 a F F , a =≥1 = 1 a. { − } ∈ ∗ ⊗ ∗ 0, , − Milnor Milnor In particular, K1(F ) = K1 (F ), K2(F ) = K2 (F ) for any field Milnor n F , and Kn (F ) is a quotient of Λ (F ×). For F an infinite field, Suslin in [24] proved that there are natural maps

KMilnor(F ) K (F ) KMilnor(F ) n → n → n whose composition is ( 1)n 1(n 1)!. This immediately implies, for exam- − − − ple, that the higher K-groups of a field of high transcendence degree are very large.

Remark 5.7. It is difficult to even briefly mention K2 of fields without also mentioning the deep and import theorem of Mekurjev and Suslin [23]: for any field F and positive integer n,

2 2 K (F )/nK (F ) H (F, µ⊗ ). 2 2 ( n 2 2 1 In particular, H (F, µn⊗ ) is generated by products of elements in H (F, µn) = µn(F ). Moreover, if F contains the nth roots of unity, then

K (F )/nK (F ) Br(F ), 2 2 ( n An Introduction to K-theory 59

where nBr(F ) denotes the subgroup of the of F consisting of elements which are n-torsion. In particular, nBr(F ) is generated by “cyclic central simple algebras”. The most famous success of K-theory in recent years is the following theorem of Voevodsky [26], establishing a result conjectured by Milnor. Theorem 5.8. Let F be a field of = 2. Let W (F ) denote the , Witt ring of F , the quotient of the Grothendieck group of symmetric inner product spaces modulo the ideal generated by the hyperbolic space 1 1 * + ⊕ *− + and let I = ker W (F ) Z/2 be given by sending a symmetric inner { → } product space to its rank (modulo 2). Then the map n KMilnor(F )/2 KMilnor(F ) In/In+1, a , . . . , a ( a 1) n · n → { 1 n} 0→ * i+ − .i=1 is an isomorphism for all n 0. Here, a is the 1 dimensional symmetric ≥ * + inner product space with inner product ( , ) defined by (c, d) = acd. − − a a Suslin also proved the following theorem, the first confirmation of a series of conjectures which now seem to be on the verge of being settled. Theorem 5.9. Let F be an algebraic closed field. If F has characteristic 0 and i > 0, then K2i(F ) is a Q vector space and K2i 1(F ) is a direct sum of − Q/Z and a rational vector space. If F has characteristic p > 0 and i > 0, then K2i(F ) is a Q vector space and K2i 1(F ) is a direct sum of "=pQ"/Z" − ⊕ , and a rational vector space. Question 5.10. What information is reflected in the uncountable vector spaces Kn(C) Q? Are there interesting structures to be obtained from ⊗ these vector spaces?

5.4 Negative K-groups Bass introduced negative algebraic K-groups, groups which vanish for regular rings or, more generally, smooth varieties. These negative K-groups measure the failure of K-theory in positive degree to obey “homotopy invariance” and “localization” (i.e.,

? 1 ? 1 K (X) = K (X A ), K (X) K 1(X) = K (X A 0 ). ∗ ∗ × ∗ ⊕ ∗− ∗ × \{ } Very recently, there has been important progress in computing these negative K-groups by Cortinas, Haesemeyer, Schlicting, and Weibel. 60 E.M. Friedlander

Question 5.11. Can negative K-groups give useful invariants for the geo- metric study of singularities?

5.5 Algebraic versus topological vector bundles Let X be a complex projective variety, and let Xan denote the topological space of complex points of X equipped with the analytic topology. Then any algebraic vector bundle E X naturally determines a topological vector → bundle Ean Xan. This determines a natural map → K (X) K0 (Xan). 0 → top Challenge 5.12. Understand the kernel and image of the above map, espe- cially after tensoring with Q:

0 an ev an CH∗(X) Q K0(X) Q K (X ) H (X , Q). (3) ⊗ ( ⊗ → top ⊗ (

The kernel of (3) can be identified with the subspace of CH∗(X) Q ⊗ consisting of rational equivalence classes of algebraic cycles on X which are homologically equivalent to 0. an The image of (3) can be identified with those classes in H∗(X , Q) rep- resented by algebraic cycles – the subject of the !

In positive degree, the analogue of our map is uninteresting.

Proposition 5.13. If X is a complex projective variety, then the natural map i an Ki(X) Q K− (X ), i > 0 ⊗ → top is the 0-map.

5.6 K-theory with finite coefficients Although the map in positive degrees

i an K (X) K− (X ) i → top is typically of little interest, the situation changes drastically if we consider K-theory mod-n. As an example, we give the following special case of a theorem of Suslin. Recall that (Spec C)an is a point, which we denote by +. An Introduction to K-theory 61

Theorem 5.14. (cf. [25]) The map i Ki(Spec C) K− (+) → top is the 0-map for i > 0. On the other hand, for any positive integer n and any integer i 0, the map ≥ i Ki(Spec C, Z/n) K− (+, Z/n) → top is an isomorphism. How can the preceding theorem be possibly correct? The point is that K2i 1(Spec C) is a divisible group with torsion subgroup Q/Z. Then, we see − that this Q/Z in odd degree integral homotopy determines a Z/n in even degree mod-n homotopy. This is exactly what Kt−op∗(+) determines in even mod-n homotopy degree. The K-groups modulo n are defined to be the homotopy groups modulo n of the K-theory space (or spectrum). Definition 5.15. For positive integers i, n > 1, let M(i, Z/n) denote the i i 1 C.W. complex obtained by attaching an i-cell D to S − via the map ∂(Di) = Si 1 Si 1 given by multiplication by n. − → − For any connected C.W. complex, we define

πi(X, Z/n) [M(i, Z/n), X], i, n > 1. ≡ If X = Ω2Y , we define

πi(X, Z/n) [M(i + 2, Z/n), Y ], i 0, n > 1. ≡ ≥ i 1 Since S − M(i, Z/n) is the cone on the multiplication by n map n → Si 1 Si 1, we have long exact sequences − → − n πi(X) πi(X) πi(X/Z/n) πi 1(X) · · · → → → → − → · · · Perhaps this is sufficient to motivate our next conjecture, which we might call the Quillen-Lichtenbaum Conjecture for smooth complex algebraic vari- eties. The special case in which X = Spec C is the theorem of Suslin quoted above. Conjecture 5.16. (Q-L for smooth C varieties) If X is a smooth complex variety of dimension d, then is the natural map

top an Ki(X, Z/n) K (X , Z/n) → i an isomorphism provided that i d 1 0? ≥ − ≥ 62 E.M. Friedlander

Remark In “low” degrees, K (X, Z/n) should be more interesting and will ∗ top not be periodic. For example, Kev (X, Z/n) has a contribution from the Brauer group of X whereas K0(X, Z/n) does not.

5.7 Etale K-theory It is natural to try to find a good “topological model” for the mod-n algebraic K-theory of varieties over fields other than the complex numbers. Suslin’s Theorem in its full generality can be formulated as follows

Theorem 5.17. If k is an algebraically closed field of characteristic p 0, ≥ then there is a natural isomorphism

et K (k, Z/n) & K (Spec k, Z/n), (n, p) = 1. ∗ → ∗

Moreover, if the characteristic of k is a positive integer p, then Ki(k, Z/p) = 0, for all i > 0.

We have stated the previous theorem in terms of etale K-theory although this is not the way Suslin formulated his theorem. We did this in order to introduce the etale topology, a associated to the etale site. For this site, the distinguished morphisms are etale morphisms E of schemes. A map of schemes f : U V is said to be etale (or “smooth → of relative dimension 0) if there exist affine open coverings U of U, V { i} { j} of V such that the restriction to Ui of f lies in some Vj and such that the corresponding map of commutative rings A R is unramified (i.e., for i ← j all homomorphisms from R to a field k, A k k is a finite separable k ⊗R ← algebra) and flat. The etale topology was introduced by Grothendieck partly to reinter- pret of fields and partly to algebraically realize singular cohomology of complex algebraic varieties. The following “comparison theo- rem” proved by and Alexander Grothendieck is an important property of the etale topology. (See, for example, [21].)

Theorem 5.18. (Artin, Grothendieck) If X is a complex algebraic variety, then an H∗ (X, Z/n) H∗ (X , Z/n). et ( sing Here, He∗t(X, Z/n) denotes the derived of the global section functor applied to the constant sheaf Z/n on the etale site. An Introduction to K-theory 63

The etale topology not only enables us to define etale cohomological groups, but also etale homotopy types. Using the etale homotopy type, etale K-theory (defined by Bill Dwyer and myself) can be defined in a manner similar to topological K-theory. For this theory, there is an Atiyah-Hirzebruch spectral sequence

p,q p q p+q E = H (X, K (+)) K (X, Z/n) 2 et et ⇒ et provided that X is a sheaf of Z[1/n]-modules. If we let µn denote the etale O q/2 j sheaf of n-th roots of unity and let µn⊗ denote µn⊗ if q = 2j and 0 if j is odd, then this spectral sequence can be rewritten

p,q p q/2 et E2 = Het(X, µ⊗ ) Kq p(X, Z/n). ⇒ − Using etale K-theory, we can reformulate and generalize the Quillen- Lichtenbaum Conjecture (originally stated for Spec K, where K is a number field), putting this conjecture in a quite general context.

Conjecture 5.19. (Quillen-Lichtenbaum) Let X be a of finite type over a field k, and assume that n is a positive integer with 1/n in k or A. Then the natural map

et Ki(X, Z/n) K (X, Z/n) → i is an isomorphism for i 1 greater or equal to the mod-n etale cohomological − dimension of X.

This conjecture appears to be proven, or near-proven, thanks to the work of Rost and Voevodsky on the Bloch-Kato Conjecture.

5.8 Integral conjectures

There has been much progress in understanding K-theory with finite coef- ficients, but much less is known about the result of tensoring the algebraic K-groups with Q. The following theorem of Soul´e is proved by investigating the group ho- mology of general linear groups over fields. Soul´e proves a vanishing theorem for more general rings R with a range depending upon the “stable range” of R. 64 E.M. Friedlander

Theorem 5.20. (Soul´e) For any field F ,

K (F )(s) = 0, s > n. n Q Here K (F )(s) is the s-eigenspace with respect to the action of the Adams n Q operations on Kn(F ). This motivates the following Beilinson-Soul´e vanishing conjecture, part of the Beilinson Conjectures discussed in the next lecture. This conjecture is now known if we replace the coefficients Z(n) by their finite coefficients analogue Z/'(n). Conjecture 5.21. (Beilinson-Soul´e) For any field F , the motivic cohomol- ogy groups Hp(Spec F, Z(n)) equal 0 for p < 0. Yet another auxillary K-theory has been developed to investigate K- theory of complex varieties, especially some aspects involving rational coef- ficients (cf. [22]). Theorem 5.22. (Friedlander-Walker) Let X be a complex quasi-projective variety. The map from the algebraic K-theory spectrum (X) to the topo- K logical K-theory spectrum (Xan) factors through the “semi-topological Ktop K-theory spectrum sst(X). K (X) sst(X) (Xan). K → K → Ktop The first map induces an isomorphism in homotopy groups modulo n, whereas the second map induces an isomorphism for certain special varieties and typ- ically induces an isomorphism after “inverting the Bott element.” This semi-topological K-theory is related to cycles modulo algebraic equivalence is much the same way as usual algebraic K-theory is related to Chow groups (cycles modulo rational equivalence). One important aspect of this semi-topological K-theory is that leads to conjectures which are “integral” whose reduction modulo n give the familiar Quillen-Lichtenbaum Conjecture. We state one precise form of such a conjecture, essentially due to Suslin. Conjecture 5.23. Let X be a smooth, quasi-projective complex variety. Then the natural map sst i an K (X) K− (X ) i → top is an isomorphism for i dim(X) 1 and a monomorphism for i = ≥ − dim(X) 2. − An Introduction to K-theory 65

Now, we also have a “good semi-topological model” for the K-theory of quasi-projective varieties over R, the real numbers. This is closely related to “Atiyah Real K-theory rather than the topological K-theory we have discussed at several points in these lectures.

Challenge 5.24. Develop a semi-topological K-theory for varieties over an arbitrary field.

5.9 K-theory and Quadratic Forms another topic of considerable interest is Hermetian K-theory in which we take into account the presence of quadratic forms. Perhaps this topic is best left to Professor Vishik!

6 Beilinson’s vision partially fulfilled

6.1 Motivation

In this lecture, we will discuss Alexander Beilinson’s vision of what algebraic K-theory should be for smooth varieties over a field k (cf. [28], [30], and [31]). In particular, we will provide some account of progress towards the solution of these conjectures. Essentially, Beilinson conjectures that algebraic K- theory can be computed using a spectral sequence of Atiyah-Hirzebruch type using “motivic complexes” Z(n) which satisfy various good properties and whose cohomology plays the role of singular cohomology in the Atiyah- Hirzebruch spectral sequence for topological K-theory. Although our goal is to describe conjectures which would begin to “ex- plain” algebraic K-theory, let me start by mentioning one (of many) reasons why algebraic K-theory is so interesting to algebraic geometers (and alge- braic number theorists). It has been known for some time that there can not be an algebraic theory whose values on complex algebraic varieties is integral (or even rational) of the associated analytic space. Indeed, Jean-Pierre Serre observed that this is not possible even for smooth projec- tive algebraic curves because some such curves have automorphism groups which do not admit a representation which would be implied by functorial- ity. On the other hand, algebraic K-theory is in some sense integral – we define it without inverting residue characteristics or considering only mod-n coefficients. Thus, if we can formulate a sensible Atiyah-Hirzebruch type 66 E.M. Friedlander

spectral sequence converging to algebraic K-theory, then the E2-term offers an algebraic formulation of integral cohomology. Before we launch into a discussion of Beilinson’s Conjectures, let us recall two results relating algebraic cycles and algebraic K-theory which precede these conjectures. The first is the theorem of Grothendieck mentioned earlier relating alge- braic K0(X) to the Chow ring of algebraic cycles modulo algebraic equiva- lence.

Theorem 6.1. If X is a smooth variety over a field k, then the Chern character determines an isomorphism

ch : K0(X) Q CH∗(X) Q. ⊗ ( ⊗ The second is Bloch’s formula proved in degree 2 by Bloch and in general by Quillen.

Theorem 6.2. Let X be a smooth variety over a field and let Ki denote the Zariski sheaf associated to the presheaf U K (U) for an open subset 0→ i U X. Then there is a convergent spectral sequence of the form ⊂ p,q p E2 = H (X, Kq) Kq p(X). Zar ⇒ − 6.2 Statement of conjectures We now state Beilinson’s conjectures and use these conjectures as a frame- work to discuss much interesting mathematics. It is worth emphasizing that one of the most important aspects of Beilinson’s conjectures is its explicit nature: Beilinson conjectures precise values for algebraic K-groups, rather than the conjectures which preceded Beilinson which required the degree to be large or certain torsion to be ignored.

Conjecture 6.3. (Beilinson’s Conjectures) For each n 0 there should ≥ be complexes Z(n), n 0 of sheaves on the Zariski site of smooth quasi- ≥ projective varieties over a field k, (Sm/k)Zar which satisfy the following:

1. Z(0) = Z, Z(1) ∗[ 1]. ( O − n Milnor 2. H (Spec F, Z(n)) = Kn (F ) for any field F finitely generated over k.

3. H2n(X, Z(n)) = CHn(X) whenever X is smooth over k. An Introduction to K-theory 67

4. Vanishing Conjecture: Z(n) is acyclic outside of [0, n]:

p H (X, Z(n)) = 0, p < 0.

5. Motivic spectral sequences for X smooth over k:

p,q p q E2 = H − (X, Z( q)) K p q(X), − ⇒ − − p,q p q E2 = H − (X, Z/'( q)) K p q(X, Z/'), if 1/' k. − ⇒ − − ∈ 6. Beilinson-Lichtenbaum Conjecture:

L n Z(n) Z/' τ nRπ µ⊗ , if 1/' k ⊗ ( ≤ ∗ " ∈ where π : etale site Zariski site is the natural “forgetful continuous → map” and τ n indicates truncation. ≤ i (n) 7. H (X, Z(n)) Q K2n i(X) . ⊗ ( − Q In other words, Beilinson conjectures that there should be a bigraded motivic cohomology groups Hp(X, Z(q)) computed as the Zariski cohomology of motivic complexes Z(q) of sheaves which satisfy good properties and are related to algebraic K-theory as singular cohomology is related to topological K-theory.

6.3 Status of Conjectures Bloch’s higher Chow groups CHq(X, n) (cf. [29]) serve as motivic cohomol- ogy groups which are known to satisfy most of the conjectures, where the correspondence of indexing is as follows:

q 2q n CH (X, n) H − (X, Z(q)). (1) ( Furthermore, Suslin and Voevodsky have formulated complexes Z(q), q 0 ≥ and Voevodsky has proved that the (hyper-)cohomology groups of these complexes satisfy the relationship to Bloch’s higher Chow groups as in (1). Presumably, these constructions will be discussed in detail in the lectures of Professor Levine. For completeness, I sketch the definitions. Recall that the standard (algebro-geometric) n-simplex ∆n over a field F (which we leave implicit) is given by Spec F [t0, . . . , tn]/Σiti = 1. 68 E.M. Friedlander

Definition 6.4. Let X be a quasi-projective variety over a field. For any q, n 0, we define zq(X, n) to be the free abelian group on the set of cycles ≥ W X ∆n of codimension q which meet all faces X ∆i X ∆n ⊂ × × ⊂ × properly. This admits the structure of a simplicial abelian group and thus a with boundary maps given by restrictions to (codimension 1) faces. The Bloch higher Chow group CHq(X, n) is defined by

q 2q n q CH (X, n) = H − (z (X, )). ∗ The values of Bloch’s higher Chow groups are “correct”, but they are not given as (hyper)-cohomology of complexes of sheaves and they are so directly defined that abstract properties for them are difficult to prove. The Suslin-Voevodsky motivic cohomology groups fit in a good formalism as en- visioned by Beilinson and agree with Bloch’s higher Chow groups as verified by Voevodsky.

Definition 6.5. Let X be a quasi-projective variety over a field. For any q 0, we define the complex of sheaves in the cdh topology (the Zariski ≥ topology suffices if X is smooth over a field of characteristic 0)

n n 1 Z(q) = C (cequi(P , 0)/cequi(P − , 0))[ 2n] ∗ − defined as the shift 2n steps to the right of the complex of sheaves whose value on a Zariski open subset U X is the complex ⊂ n j n 1 j j cequi(P , 0)(∆ )/cequi(P − )(U ∆ ) 0→ × n j n where cequi(P , 0)(U ∆ ) is the free abelian group on the cycles on P × × U ∆j which are equidimensional of relative dimension 0 over U ∆j. × × Conjecture (1) is essentially a normalization, for it specifies what Z(0) and Z(1) must be. Bloch verified Conjecture 2 (essentially, a result of Suslin), Conjecture 3, and Conjecture 7 (the latter with help from Levine) for his higher Chow groups. Bloch and Lichtenbaum produced a motivic spectral sequence for X = Spec k; this was generalized to a verification of the full Conjecture (5) by Friedlander and Suslin, and later proofs were given by Levine and then Suslin following work of Grayson. The Beilinson-Lichtenbaum conjecture in some sense “identifies” mod-' motivic cohomology in terms of etale cohomology. Suslin and Voevodsky proved that this Conjecture (6) follows from the following: An Introduction to K-theory 69

Conjecture 6.6. (Bloch-Kato Conjecture) For fields F finitely generated over k, Milnor n n K Z/' H (Spec F, µ⊗ ). n ⊗ ( et " In particular, the Galois cohomology of the field F is generated multiplica- tively by classes in degree 1. For ' = 2, the Bloch-Kato Conjecture is a form of Milnor’s Conjecture which has been proved by Voevodsky. For ' > 2, a proof of Bloch-Kato Conjecture has apparently been given by Rost and Voevodsky, although not all details have been made available. This conjecture will be the main focus of Professor Weibel’s lectures. This leaves Conjecture (4), one aspect of this is the following Vanishing Conjecture due to Beilinson and Soul´e. Conjecture 6.7. For fields F , K (F )(q) = 0, 2q p, p > 0. p Q ≤ Reindexing according to Conjecture (7), this becomes

i H (Spec F, Z(q)) = 0, i 0, q = 0. ≤ , The status of this Conjecture (4), and in particular the Beilinson-Sou´e vanishing conjecture, is up in the air. Experts are not at all convinced that this conjecture should be true for a general field F . It is known to be true for a number field.

6.4 The Meaning of the Conjectures Let us begin by looking a bit more closely at the statement

Z(1) ∗[ 1] ( O − of Conjecture (1).

Convention If C∗ is a cochain complex (i.e., the differential increases degree i i+1 by 1, d : C C ), we define the chain complex C∗[n] for any n Z as → j ∈ j the shift of C∗ “n places to the right”. In other words, (C∗[n]) = C∗− . In particular, [ 1] is the complex (of Zariski sheaves) with only one O∗ − non-zero term, the sheaf of units, placed in degree -1 (i.e., shifted 1 place O∗ to the left). In particular,

1 H∗ (X, ∗[ 1]) = H∗− (X, ∗); Zar O − Zar O 70 E.M. Friedlander thus, 1 2 P ic(X) = H (X, ∗ ) = H (X, Z(1)). Zar OX This last equality is a special case of item (3). Perhaps it would be useful to be explicit about what we mean by the cohomology of a complex C∗ of Zariski sheaves on X. A quick way to define this is as follows: find a map of complexes C I with each Ij an injective ∗ → ∗ object in the category of sheaves (an ) such that the map on cohomology sheaves is an isomorphism; in other words, for each j, the map of presheaves

j j+1 j 1 j ker d : C C /im d : C − C { → } { → } j j+1 j 1 j ker d : I I /im d : I − I → { → } { → } induces an isomorphism on associated sheaves

j j (C∗) (I∗) H ( H for each j. A fundamental property of this cohomology is the existence of “hypercohomology spectral sequences”

p,q p q p+q $E = H (X, C ) H (X, C∗) 1 ⇒ p,q q j p+q E = H (X, (C∗)) H (X, C∗) 2 H ⇒ Conjecture (2) helps to pin down motivic cohomology by specifying what the top dimensional motivic cohomology (thanks to Conjecture (4)) should be for a field. Since Milnor K-theory and algebraic K-theory of the field k are different, this difference must be reflected in the other motivic cohomology groups of the field and tied together with the spectral sequence of Conjecture (5). Conjecture (2) can be viewed as “” for it deals with subtle invariants of the field k. Conjecture (3) is “geometric”, stating that motivic cohomology reflects global geometric properties of X. Observe that since we are taking Zariski cohomology, Hn(Spec k, ) = 0 for n > 0 and this item − simply says that CH0(Spec k) = Z, CHn(Spec k) = 0, n > 0. Bloch has also proved that the spectral sequence of Conjecture (5) col- lapses after tensoring with Q; indeed, Conjecture (7) proved by Bloch is a refinement of this “rational collapse”. Conjectures (3) and (5) together with this collapsing gives Grothendieck’s isomorphism K(X)Q CH∗(X). By ( An Introduction to K-theory 71 simply re-indexing, one can write the spectral sequence of Conjecture (5) in the more familiar “Atiyah-Hirzebruch manner”

p,q p E2 = H (X, Z( q/2)) K p q(X) − ⇒ − − where Z( q/2) = 0 if q is not an even non-positive integer and Z( q/2) = − − Z(i) is q = 2i 0. − ≥ Let me try to “draw” this spectral sequence, using the notation

(q) i Kq i H (X, Z(q) − ≡ as in Conjecture (7).

Z

0 P ic(X) O∗

(2) (2) 2 0? K2 K1 CH (X)

(3) (3) (3) 3 0? K3 K2 K1 CH (X)

(4) (4) (4) (4) 4 0? K4 K3 K2 K1 CH (X)

In this picture, the associated graded of K0 is given by the right-most diag- onal, then gr(K1) by the next diagonal to the left, etc. The top horizontal row is the “weight 0” part of K , the next row down is the “weight 1” part ∗ of K , etc. There is conjectured vanishing at and to the left of the positions ∗ with 0? in the picture – i.e., to the left.

6.5 Etale cohomology Our final task is to introduce the etale topology and attempt to give some understanding why Conjecture (6) of the Beilinson Conjectures comparing mod-' motivic cohomology with mod-' etale cohomology makes motivic co- homology more understandable. 72 E.M. Friedlander

Grothendieck had the insight to realize that one could formulate sheaves and in a setting more general than that of topological spaces. What is essential in sheaf theory is the notion of a covering, but such a covering need not consist of open subsets.

Definition 6.8. A (Grothendieck) site is the data of a category /X of C schemes over a given scheme X which is closed under fiber products and a distinguished class of morphisms (e.g., Zariski open embeddings; or etale morphisms) closed under composition, base change and including all iso- morphisms. A covering of an object Y /X for this site is a family of ∈ C distinguished morphisms g : U Y with the property that Y = g (U ). { i i → } ∪i i i The data of the site /X together with its associated family of coverings C is called a Grothendieck topology on X. The reader is referred to [33] for a foundational treatment of etale coho- mology and to [21] for an overview.

Example 6.9. Recall that a map f : U X of schemes is said to be etale if → it is flat, unramified, and locally of finite type. Thus, open immersions and covering space maps are examples of etale morphisms. If f : U X is etale, → then for each point u U there exist affine open neighborhoods SpecA U ∈ ⊂ of u and SpecR X of f(u) so that A is isomorphic to (R[t]/g(t)) for some ⊂ h monic polynomial g(t) and some h so that g (t) (R[t]/g(t)) is invertible. $ ∈ h The (small) etale site X has objects which are etale morphisms Y X et → and coverings U Y consist of families of etale maps the union of whose { i → } images equals Y . The big etale site X has objects Y X which are ET → locally of finite type over X and coverings U Y defined as for X { i → } et consisting of families of etale maps the union of whose images equals Y . If k is a field, we shall also consider the site (Sm/k)et which is the full subcategory of (Spec k)ET consisting of smooth, quasi-projective varieties Y over k. An instructive example is that of X = SpecF for some field F . Then an etale map Y X with Y connected is of the form SpecE SpecF , where → → E/F is a finite separable field extension.

Definition 6.10. A presheaf sets (respectively, groups, abelian groups, rings, etc) on a site /X is a contravariant functor from /X to (sets) (resp., C C to groups, abelian groups, rings, etc). A presheaf P : ( /X)op (sets) is C → said to be a sheaf if for every covering U Y in /X the following { i → } C An Introduction to K-theory 73 sequence is exact:

P (Y ) P (U ) → P (U U ). → i → i ×X j .i .i,j (Similarly, for presheaves of groups, abelian presheaves, etc.) In other words, if for every Y , the data of a section s P (Y ) is equivalent to the data of ∈ sections s P (U ) which are compatible in the sense that the restrictions i ∈ i of s , s to U U are equal. i j i ×X j The category of abelian sheaves on a Grothendieck site /X is an abelian C category with enough injectives, so that we can define sheaf cohomology in the usual way. If F : /X)op (Ab) is an abelian sheaf, then we define C → i i H (X /X , F ) = R Γ(X, F ). C Etale cohomology has various important properties. We mention two in the following theorem.

Theorem 6.11. Let X be a quasi-projective, complex variety. Then the etale cohomology of X with coefficients in (constant) sheaf Z/n, H∗(Xet, Z/n), is naturally isomorphic to the singular cohomology of Xan,

an H∗(Xet, Z/n) H∗ (X , Z/n). ( sing Let X = Speck, the spectrum of a field. Then an abelian sheaf on X for the etale topology is in natural 1-1 correspondence with a (continuous) for the Gal(k/k). Moreover, the etale cohomology of X with coefficients in such a sheaf F is equivalent to the Galois cohomology of the associated Galois module,

H∗(k , F ) H∗(Gal(F /F ), F (k)). et ( From the point of view of sheaf theory, the essence of a continuous map g : S T of topological spaces is a mapping from the category of open → subsets of T to the open subsets of S. In the context of Grothendieck , we consider a map of sites g : /X /Y , a functor from /Y C → D C to cC/X which takes distinguished morphisms to distinguished morphisms. In particular, for example, Conjecture (6) of Beilinson’s Conjectures involves the map of sites

π : X X , (U X) U X. et → Zar ⊂ 0→ → 74 E.M. Friedlander

Such a map of sites induces a map on sheaf cohomology: if F : ( /Y )op D → (Ab) is an abelian sheaf on /Y , then we obtain a map C

H∗(Y /Y , F ) H∗(X /X , g∗F ). D → C 6.6 Voevodsky’s sites We briefly mention two Grothendieck sites introduced by Voevodsky which are central to his approach to motivic cohomology. The reader can find details in [34].

Definition 6.12. The Nisnevich site on smooth quasi-projective varieties over a field k, (Sm/k) , is determined by specifying that a covering U Nis { i → U of some U (Sm/k) is an etale covering with the property that for each } ∈ point x U there exists some i and some point u˜ U such that the induced ∈ ∈ i map on residue fields k(u) k(u˜) is an isomorphism. → Definition 6.13. The cdh (or completely decomposed, homotopy) site on smooth quasi-projective varieties over a field k, (Sm/k)cdh, is determined as the site whose coverings of a smooth variety X are generated by Nisnevich coverings of X and coverings Y X, X X consisting of a closed { → $ → } immersion i : Y X and a proper map g : X X with the property that → $ → the restriction of g to g 1(X i(Y )) is an isomorphism. − \ An Introduction to K-theory 75

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